 Hello friends, so welcome to this session on quadratic equation and We will be trying to revise So welcome to this We'll be trying to revise the concepts of quadratic equations in two hours time. We have some slides prepared for you and So to begin with I'll need who all are present so I can see the few students who are already there So let's begin our class. So quadratic equation is an important chapter In in your 10th board portion. So just to check whether you are able to hear my Voice you can confirm, you know, if you are able to hear my voice in the chat Okay, so just a confirmation if you're able to hear my so anyone can confirm that, you know, you're able to Visualize and are you able to see the screen? Okay, good. Is the screen visible? Okay, so let's start and The program is something like this We will be revising all the concepts which are there in this chapter and then towards the end We will try and solve as many questions as possible depending upon the time which we'll be having So, you know, let's begin. So basically we start with polynomial, right? So what is a quadratic polynomial? So a quadratic polynomial is nothing but a polynomial of degree 2 degree 2 And it's called a quadratic polynomial. So any polynomial with degree 2 is called a quadratic polynomial The general form of a quadratic polynomial is a x square. So hence general form of a quadratic polynomial is a x square plus b x plus c where a, b, c are real numbers, right? So a, b and c all belong to the set of real numbers and A cannot be 0, right? If a becomes 0 then this polynomial is reduced to a linear polynomial So hence a cannot be 0. So this is an example or let's say this is a definition of a quadratic polynomial Yeah, okay. So what's next? So let us go to the next slide We'll try to cover the, you know, topics quickly and then we will be going to Problem solving. So now the second thing is If px equals a x square plus b x plus c and a not equal to 0 is a quadratic polynomial And alpha is a real number then p of alpha is equal to a alpha square plus b alpha plus c Is known as the value of the quadratic polynomial, right? So hence example will be let's say px is Equal to 2x square plus x plus 1 let's say then p of 1 is called the value of the polynomial at x equals to 1. So at p equals to 1 will be nothing but 2 times 1 square plus 1 plus 1 which is nothing but 4, right? So hence this is meant by What is meant by the value of the polynomial at any given x, okay? Next In case during this, you know, we will be a little Fast in revising. So in case you have any doubt you can always post either as a whatsapp query or In the chat sessions also. Yeah next is Next is a real number alphas is said to be a zero, right a near a real number again We are dealing with real numbers a real number alpha is said to be a Zero of the quadratic polynomial, right? So let's say if I have a quadratic polynomial px equals ax square plus bx plus c, please remember always this is the general form of the quadratic polynomial And if alpha is the zero If alpha is the zero then that is if alpha is a zero a zero of this of px then Alpha is that value which will make px zero. So hence that means if I if I Plug in x plug in alpha in place of x You will get p of alpha zero. That means a times alpha square Plus b alpha plus c equals zero, right? So let's take an example if px is let's say a square minus 2x or rather We will be taking it as x square So x square minus 2x plus 1, okay? So if you see p of 1 that means if I plug in 1 in place of x I will get 1 square minus 2 times 1 plus 1 Which is 0. So hence we can say 0 or we can say 1 is the zero zero of px Okay, this is What this is meant this is meant by what is meant by zero of a polynomial, okay? Next moves to the next slide next slide is Yeah, so next one is card number four which says if px Is equal to ax square plus bx plus c is a quadratic polynomial Then px equals to zero that is so this is definition of a quadratic equation So equate any quadratic polynomial to zero You will get a quadratic quadratic equation. There is nothing much to delve into it So quadratic equation definition is there is a polynomial which is ax square plus bx plus c So ax square Plus bx plus c let's say if you equate it to zero or for that matter any other number if you also equate it to And then express in this form ax square plus bx plus c Equals to zero then it is a quadratic equation mind you a cannot be zero a cannot be Zero why if a a becomes zero then it is reduced to bx plus c equals zero Which is nothing but a linear equation in one variable the linear equation Okay, so please be mindful of this Let's go to the next slide next slide what is there in the next slide next slide says a real number alpha A real number alpha right a real number alpha is said to be a root mind the word So if real number alpha is said to be a root Of the quadratic equation ax square plus bx plus c equals to zero If a alpha square plus b alpha plus c equals zero this is you know time and again you have done it So no point investing more time into it. So means if alpha is the root Then if you deploy that Alpha in the equation you will get both sides lhs as well as rhs to be equal to zero right A real number alpha is said to be a root of the quadratic equation if and only if Alpha is zero of the polynomial. So these are you know, uh, you know, we are trying to Relate two things. So let's say we studied zero of a polynomial Zero of a polynomial What is the zero of a polynomial that particular value of the variable which makes the polynomial zero So if zero of a polynomial will become Nothing but root of the equation root of the equation which equation If you equate let's say the polynomial was px then if you equate that px to zero Then that becomes root of root of This polynomial right root of px. So same value. Let's say alpha if alpha is zero of polynomial px Then alpha will be the root of equation px equals to zero Okay, so this is about root and zeros. Let's go to the next slide Next slide is Yeah, so Okay, so this is if a x square plus bx plus c A not equals to zero is factorizable into a product of Two linear factors then the roots of the quadratic equation can be found out by equating each factor to zero So this is nothing but now we are we are heading towards how to solve or sorry, wait a minute Okay. Yeah, so it's nothing but how to solve How to solve a quadratic A quadratic equation How to solve a quadratic equation? So method number one is factorization method factorization Factorization method. So in in your exams, there will be categorical questions where they will be saying Using factorization methods solve this quadratic equation and we'll see problems related to that a little later But uh, or we can also parallely solve, uh, you know questions like that But then the question will be factorize using factorization method solving another is something called completing the square method completing the Squire method Square method. This is the second method through which you can solve any quadratic equation and third is Third is my quadratic formula quadratic formula, right which actually quadratic Quadratic formula now quadratic formula is a direct fallout of completing the square method only so it is they are linked So quadratic formula comes from this but then now we have a direct formula to Solve any quadratic equation Correct. Now we'll take up factorization method one hour and completing the square method and quadratic formula one by one So what is factorization method? Let me Take, you know to this slide now quadratic, uh, let's say factorization method So what do you do in factorization method? factorization method Okay, so let's say you have and we'll we'll use examples. Let's say one question is given and question is Question is using quadratic. Let's factorization method solve this very simple question x square plus 6x plus 5 Equals zero will take uh, you know difficult ones as well So x square plus 6x plus 5 plus 2 zero in factorization method, which you have already learned b We try and split the middle term Right. So split the middle term such that let's say if you have a general equation a x square plus bx plus c equals zero What do you need to do? You first multiply a and c find the value Okay, and then try to break b into b 1 plus b to a sum of b 1 plus b 2 Such that b 1 b 2 the product of b 1 and b 2 should be equal to a and c a c right? This is how this is how we do the Uh, factorization method solution Okay, now So x square plus 6x now What is a here value of a let me write is 1 value of b is 6 and c is 5 So ac is equal to 5 Correct now b is equal to 6 Now I have to break 6 into two terms into two parts such that b 1 plus b 2 is 6 And b 1 b 2 must be equal to ac which is 5 now a and c are integers Okay, and 5 is a prime number So the only way you can factorize 5 is you can write 5 into 1 right? So hence b 1 becomes 5 and b 2 becomes 1 if you see it clearly satisfies This relationship so hence from here we can say It can be written as x square plus 5x plus x plus 5 equals zero So hence x times x plus 5 take x common plus 1 times x plus 5 equals zero So hence it is x plus 5 times x plus 1 equals zero now there are two two Two product there are two factors whose multiplication of product is zero So it is possible only when x plus 5 equals to zero Or x plus 1 equals to zero that means If this is true, then x equals negative 5 and if this is true then x equals negative 1 Okay, so this is the solution so hence x equals to negative 5 and x equals to negative 1 Are the solution okay? You also remember guys for quadratic equations the maximum number of solution you can have Or maximum number of Roots you can have is two either the quadratic equation will have no solution Or it will have only one solution which we say as equal solutions or it will have Two real solutions so hence what do what do I say the quadratic equation will have no real solution We'll see later on. What are the condition no real solution? No real solution or it can have two equal solutions two equal solutions and three is third is Unique or too unequal equal Solution now we are talking in terms of only real numbers. We are not going into the realm of complex numbers. So all are So if these are valid, then these are valid for as the solutions are in real domain Okay, so we will see uh some more problems on this. Okay, let's see What else let's take up some more questions on factorization method then we will proceed Okay, so the question is factorize solve the following quadratic equation by factorization method and the question is this question is x square Plus two root two root two x minus six equals zero again. What is a? A is one. What is b? b is positive two root two and c is negative six Okay, now I have to what is ac ac is six Okay, and b I have to spit in such a way that b one plus b two is equal to two root two and and b one b two is equal to six Is it a b one? Sorry not six negative six Okay, now if you see, uh, there is an irrational irrational term here Two root two is an irrational number and but b one b two is you know, which is equal to ac. Sorry ac is negative six here So negative six right or ac is or this is equal to ac. Sorry. This is equal to ac Right now we have to split in in such a way So we have to split two in such a way that the product is rational Now how can that be possible that a sum of two numbers is irrational But product is rational that can be possible only when You eliminate the irrational part from here and that is possible only when I multiply two root two by at least root two But now the sum must be yeah, the sum must be two root two But the product must be negative six now another thing to be taken care of is if the sign of b one plus b two is not same as B one b two then what do we do is we We we express b one and b two as subtraction of a Larger number minus a smaller number. So hence two root two if you see can be expressed as three root two minus root two And why did I do this? So if you multiply these two number you will certainly get six negative six rather and The sum also is two root two. So hence the splitting of the middle term in this case will be three root two x Three root two x minus root two x Minus six equals zero. So x is common to it. So x plus three root two And then here again, if you see this is minus root two common here. So this is x plus Three root two x plus three root two. This is equal to zero now if you if you See this is nothing but x plus three root two Into x minus three x minus root two equals zero So hence if you equate this, this is x plus three root two equals to zero And x minus root two equals to zero. So hence from the first one you get x equals to negative three root two And here you let x equals to root two. Okay, this is how you have to solve. Let us take a A little bit more. Yeah, so let us take a cbsc 2013 cbsc 2013 question. What is cbsc 2013 question? What does it say? Okay So let us say Let us say we are we are trying to solve cbsc cbsc 2013 question Okay, send cbsc 2013 question the question says solve again by factorization method and The equation given is one upon x minus two plus Plus plus one two upon x minus one Two upon x minus one is equal to six upon x This is cbsc 2013 some question asked in cbsc 2013 examination So one upon x minus two plus two upon x minus one equals to six by x Now clearly this is not a quadratic equation in this form, but it can be reduced to one how So let us take lcm and try and reduce it to a quadratic form So hence it is nothing but x minus two and x minus one as a common denominator So here is one so one times x minus one and here is two so two times x minus two And this must be equal to six upon x on the right hand side now The problem is reduced only for simplification. So if you simplify This is x minus one plus twice of x minus four divided by if you see this is x square minus two x minus x Plus two and this is equal to six upon x Okay, now let us cross multiply cross multiplying. I will get I will get x times x. So this is nothing but if you simplify further, this is x plus two x which is three x And minus one minus four is minus five Okay, and this has This has to be equal to six times here if you simplify you will get x squared Minus three x plus two Correct. So if you see simplify further, this is three x squared minus five x And this equals six x squared minus 18 x plus 12 Okay, now it is uh again You know, we just need to reduce it further and then you will get a quadratic equation So if you simplify everything you'll get six x square minus three x square is three x squared minus 18 x And then minus five x go on to the other side. It becomes Minus 13 x and here is plus 12 and this is equal to zero Okay, so hence we get an equation We we get an equation three x square minus 13 x Plus 12 equals to zero now again. What is a here? a is equal to three v equals to negative 13 and c equals 12 So ac is nothing but 36 Now I have to split b in such a way that v1 plus b2 Is negative 13 and v1 v2 is equal to ac which is equal to 36 Now how to do that in such cases we We we find out all the integral pair of factors of ac. So hence I'll do this work here So let's say 36 can be expressed as one into 36 Okay, clearly One and 36 will not yield you 13. So basically negative one into negative 36 will have to Do like that and then 36 can also be written as two into 18 Right, but 18 and two also will not give you 13 here. So 36 can also be written as three into 12 Okay, but this again is not going to yield minus 13 So 36 now is nothing but four into nine And this is something which is interesting. Why because nine plus four is 13. So hence we now Get the split. So hence basically it can be written as three x square minus 9x minus 4x plus 12 equals zero if you now see negative nine into negative four is 36 and negative nine plus negative four is negative 13 So hence it is now you have to just the question remains to just find out the comments So three x is gone. So it is x minus three and then here four is common for x minus three equals zero So this is reduced to x minus three times three x minus four equals zero So hence either x minus three equals zero or or three x minus four equals zero So hence from this you will get x equals to three and from this you get x equals four upon three Okay, so this is a trick for example If you see here v was negative and a and c were positive in such in such case We rest assured that The the the roots will be positive. So hence we you know, you can you can you can take down as a trick So let's say a was positive a was greater than zero v was less than zero and c was again greater than zero in such cases Both, uh, you know, uh, either both the roots will be positive or both the roots will be negative You can you can you can take that as a trick now Just to check whether you have solved the equation correctly now having seen this let's go back to our this thing So we now learned What was our factorization method? So this is done. We will see completing the square method quadratic formula in a little while now Let's go back to next slide. So yeah The roots of a quadratic equation can also be found by using the method of completing the square So let us let us do the completing the square method quickly though the proves are not asked in the exam But you must know because the process is important now. Let's go to completing the square method. Okay Okay, now, so let me So what is the completing the square method? Let's see So completing the square method, let us say So we are now dealing with completing completing the square square method so as the name suggests We need to complete some square. What kind of a square here? It is a binomial square We are going to Complete how let us say the equation was a x square plus b x plus c equals zero Where a clearly is not equal to zero Now what we can do is divide the equation divide the equation By a So what will you get you will get x square plus b upon ax Plus c upon a equals zero. Is it? Now if you see guys, there is a square term And there is a term which contains x and another another factor and there's a constant, right? So what we can Do is we can try and complete this square means what? Let's say this is half a rectangle I have shown why because we have to yet to complete this square. What does it mean? It means that if I somehow get a form x plus alpha square What is the form if you expand it you'll get x square plus 2 alpha x plus Alpha square, isn't it? So if you see these two terms resemble the first two terms of this Expansion of a square so can be hence if we somehow add alpha square Then we can complete the whole square that that is what it means. So let us see how we do it So hence I'm saying it is x square plus b upon ax plus c upon a equals zero Now if you see I can say this is x square plus since I need a factor of two here So then what can what we can do is we can multiply and divide by Two both so basically we multiply and divide this term by two So hence I can say this as two into x Into so this x and this I have added an extra two here, which I will take out by doing this b upon weight Right if you see now this whole term This whole term Is nothing but b by ax But I have just manipulated it to write like that. Okay Now this gives you a sense of x square plus two a b kind of thing, right? So where this is b so hence I can Very well, right b by two a Whole square, isn't it? But then if I have added something extra to the equation I can get back to the original equation by subtracting the same quantity, isn't it? So this I can do now this addition and subtraction nullifies the effect And then finally I can add c by a which was already there Now why did I do that if you see you can clearly see a pattern here This is uh, this is a square, right? So hence if you see hence if you see Hence if you see this is nothing but x plus b upon two a Whole square, right x square plus two x times b by two a plus b by two a whole square can be expressed as x plus b upon two a whole square And this is equal to I can take all these extra terms to the right hand side And I can say this is b square by four a square plus c upon a Isn't it now I am writing it here I am writing it here so that I don't lose the continuity now if you see from here I can say x plus b upon two a whole square Is equal to b square plus if you take the lcm and you know Simplify it it will be four ac by four a square Okay, so that means x plus b upon two a will be nothing but plus minus under root b square plus four a c upon four a square So hence simplifying you can say x is equal to minus b upon two a Plus minus. This is nothing but under root b square plus four ac upon twice of a Right because under root four a square will come out as twice of a so hence it is nothing but minus b plus minus under root b square plus four ac upon twice of a Right. So this this completing the square method itself leads to let's say, you know Solving the quadratic or finding the quadratic formula if you see this is nothing but the quadratic formula quadratic formula quadratic formula there Where we now know that x directly I can find out by deploying all the values here in terms of a b and c And you can directly find x Oh, I see. I'm really sorry. Yeah, there is some some error This is when I take c by a it is it is it is sorry for This error. This is not plus. This is minus. Thanks for correcting So this is clearly minus. So hence, uh, here it is minus. I'm sorry So all the pluses plus c will be minus c. Yeah, so that's the correction. Thanks for correction. So this is plus Plus plus everywhere it is minus minus minus Minus and minus. Yeah, so minus b plus minus. Thanks for correcting minus b plus minus Under root b square minus four ac by twice of a that is what is quadratic formula So if you see there are two roots possible there are, you know, if you see there are plus and minus There are two roots possible one is called. Let's say alpha. So alpha is minus b plus of under root b square minus four ac upon twice of a And beta is minus b minus under root b square minus four ac upon twice of a These are the two These are the two Okay, these are the two roots. Let's say if we have some previous year question on solving the quadratic equation by completing the square So, yes, so it is Okay, so there are lots of ncrd questions, but we will pick up something which has been asked in the previous year So that okay, so Okay, never mind. So let's solve this Okay, the question is solved by method of completing the square. What is the question? question is Yeah, solve by completing the square So the question says this 4x square plus 4x square plus 3x 4x square plus 3x plus 5 Plus 5 equals 0 again 4x square plus 3x plus 5 equals to 0 Okay, so first step is divided by a so hence it will be x square plus 3 upon 4x plus 5 upon 4 equals 0 Okay, so now what do I do x square plus twice of x into Three by eight. Why because I have to multiply two in the denominator Okay, but since and then the b part here is three by eight So hence you have to write three by eight whole square But then as you added something you have to subtract also so three by eight holes fired And then here it is plus 5 upon 4 which is equal to 0 Okay, so hence what do I get I get x plus 3 by 8 whole squared Is equal to 3 by 8 whole squared minus 5 upon 4 minus 5 upon 4. So let us simplify this. This is 9 upon 64 Right minus 5 upon 4 Which is nothing but 64 will be the LCM And hence it is 9 and hence it is hence it is Hence it is 9 minus 16 4 just 64 so 16 5 is the 80. So here it is negative 71 by 64 Now here is the problem. The problem is that there is a square term on the LHS. So LHS is greater than zero. Why because squares are always either zero or let's say positive value, but RHS RHS is less than zero. It's a negative number. So this cannot This is not possible. So hence we say there is no real root There is no real root Of this Equation Okay, so there is no real root to this equation Then so this is what is called completing the square method guys. So this is Ticked up now next is quadratic formula. So we saw quadratic formula hand in hand and now please remember the the quadratic formula is what? minus b plus minus under root b square minus 4 ac upon twice of a Now b square minus 4 ac guys is called discriminant What is it called? This is called D and D is discriminant discriminant Okay, we will be using this thing a little later. Just keep this in mind right what is meant by D and how we use D for Various or let's say finding out the nature of roots Okay Let us take up one another good problem on Company using solving the quadratic equation using completing the square method. Okay, so so that you will get a little bit of more Hands on okay now So the question is what is the question? question says find the roots of the equation a square x square minus 3 ab x 3 ab x 3 ab x plus 2 b squared Plus 2 b square is equal to 0. This is a yeah, there are no numerals here. It is a purely, you know, lots of Variable constants so a square x square minus 3 ab x plus 2 b square equals to 0 And we will be solving it by completing the square method now again What do we do we divide the entire here in this case the coefficient of x is a square So let us divide the entire equation by a square. So dividing dividing the equation The equation by a square you will get x squared minus 3 ab x upon a squared plus Twice of b square upon sorry a squared and this is equal to 0. I'll write it a little Clearly so that avoid overwriting so that you don't Get confused yourself and you know It's uh, it's always a better practice to write as neatly as possible because you yourself should not be confused So x square minus 3 ab x by a square plus 2 b square by a square equals to 0 So hence I can say this is nothing but let's you know, so hence x square Uh, I have to now do what multiply by 2 and keep x here So that 2 x term is here and the rest of the term is 3 ab So one a will be cancelled if you see one a will get cancelled by this So hence I will write 3 3 b By a or in fact 2 a because this one 2 a over here So I'll have to add to or multiply denominator by 2 as well. So 3 b by twice of a Then now this becomes my b term So hence I'll have to complete this square. So hence I can write 3 b by twice of a Whole square, right? So if you do this the square gets completed But since you have added an extra term to the equation, you have to remove it from it as well So 3 b by 2 a whole square Plus 2 b square by a square and this is equal to 0 So hence what can I say from this thing? So this is x minus 3 b upon twice of a whole square Because if you see This is the expansion, isn't it? This is the expansion of this first three term First three terms these three terms here Is the square term, right? And now this must be equated to So take everything on the other side, you'll get 9 b square So 3 b square is 9 b square and then minus sign becomes positive It's 4 b square and then here it is negative 2 b square by a square I'm sorry. This is not b square in the denominator. This is a square This is a square. Okay, so Yeah a square I'll write it properly so that you don't get confused. You should also avoid Overwriting. Yeah, it's 9 b square by 4 a square minus 2 b square by a square. So simplify this what do you get 4 a square is the lcm So hence it is 9 b square and a square and multiplied by 2. So it is 8 b square So it is b square by 4 b square by 4 a square Yeah, so now if you remove the square sign you take what do square root on both sides You'll get x minus 3 b upon twice of a Is equal to plus minus b upon 2 a Okay, so hence what do I get a solution so x is nothing but 3 b upon twice of a Plus minus b upon twice of a So which is nothing but so x alpha is nothing but 3 b plus b upon twice of a which is 4 b by twice of a which is 2 b by a This is solution number one and solution number 2 beta is equal to 3 b by twice of a Minus b upon twice of a Which is nothing but 3 b minus b upon twice of a which is nothing but b upon a So if you see the equations to this, sorry the solutions to this equation would be x equals 2 b by a And y equals b by a now Please remember whenever you have such equations in in the examination paper always put a star mark against that question because You need to spare some time towards the end of the question paper as we have discussed multiple times To check the solution. So while you check the solution You can put any of these if let's say if you have lesser time to revise Put any of these solutions which one which you think is a simpler one and see whether it went while Plugging into the equation you get zero or not that will be a good enough indication whether you have solved it correctly Yes, if you have enough time you can go through the entire solution once again So use your judgment during the exam if you have lesser time Plug in the solution and see whether You have solved it correctly or not. So but During the rush hour that means towards the last 10 minutes You will not be knowing which question was to be revised once again So hence I told you that put a mark in the In the answer script somewhere or you can write down towards the end that these questions need to be revised once again Okay, fair enough. So let's move on to our next discussion. So here is so we discussed this roots of a quadratic equation Can also be found by using the method of completing the square And next is this quadratic formula now here is Important things which is anyways is discussed in the next slide. So let me go to the next slide now Here is where we are talking about nature of roots of quadratic equation Okay, so, you know, there will be questions on nature of roots of quadratic equation. Yeah, so How do we find out? So if you see in our quadratic formula Let me just delete this so we are now dealing with quadratic formula as I told you previously We will come back to the discriminant thing and here is what I meant. So let's say Let's say your quadratic formula alpha was equal to minus b plus under root d upon twice of a is it And beta was also equal to minus b minus under root d upon twice of a Right, these are the two Solution or two roots of the quadratic equation ax square plus bx plus c equals to zero now The problem is if d what was d d is nothing but b square minus four of ac Right now there is a root of d. So hence d must be greater than equal to zero For alpha and beta to be real Is it it? Why because there is no solution to you know, square root of negative value in real number set So hence d must be greater than equal to zero if d is less than zero then we say that there are non real solutions Right non real non real solution Those are those lines in the realm of complex numbers But in your cbsc 10th grade portions, we are not going to deal with them So hence if d is less than zero then there will be non real solution and if d is equal to zero That means d equals to zero then both the roots are equal, isn't it? So if you see alpha is also equal to minus b upon two a And beta is also equal to minus b upon two a so hence in this case we say Equal real and equal roots real roots are definitely real, but they are equal They are equal and third is d is greater than greater than Zero in this case you will have two distinct values of alpha and beta and we say that equations have real and distinct or real and different roots to Real and distinct this is the term distinct Roots so you will get two roots which are different from each other So please remember these three conditions questions will be asked on these the question will be you know to find out some value of a b and c given that The roots are either distinct and real equal real and unequal or real and equal or Not real imagine right this can be question. We'll see such questions later on. Okay. Now. Let's go to the next slide Okay, so this summarized, you know, so we kind of you know, these are the Theory portions related to quadratic equation now the questions would be either Of you know, let's discuss what type of questions would be could be asked So in our experience in previous year papers also you have seen the questions asked are either of This form they will give you solving the equation Solving the equation Okay, usually it is of two and three marks So solving the equation will be one right and now they will ask you categorically So, please be very very careful while you are solving the equation as in what kind of method they are asking So if you see in the previous slide, I'll show you So if you see here in the next slide, they are saying Solve by factorization method. We just solved this question. This is actually a cvc question So it is categorically saying that solve by factorization method So when and be very very careful whenever, you know during excitement What happens people think that we know how to solve quadratic equation and they miss on the Instruction given on the question paper, right? So don't miss on the instruction given on the question paper when it says factorization method You cannot apply quadratic formula and When it is, you know, when it's nothing is mentioned, then it is your choice Whether you can use your factorization method you want to complete the square Or you want to use quadratic formula most of the questions which will be there in the cvc board paper They will be, you know, kind of You know, then you can solve it by factorization method mostly most of them Okay, so hence it's always good to solve through factorization method because it is, you know, simpler also and lesser Pro calculation mistakes But yes, it is always a good practice to come back and check it using our quadratic formula whether you have got the correct You know, I'm saying in the in the rough side You can always do a quickly, you know quick check and then see whether the solution which you have a right at Is correct or not, right? See all the theory which we discussed you would be knowing it already now the problem would be when you'll be solving the questions Either you will misread the question and hence lose marks or while solving it you will make careless mistakes and then again miss marks And then as we were discussing these were the types of question Which will be asked especially in the word problem if you misinterpret the question Then you might lose mark. So hence, please be very very Careful about it. So hence what type of questions I talk I told you either it will be solving the equation so hence be careful what is the method is what is the method which is being asked whether it is a factorization method So you have to the moment it is a factorization method you have to Use the spreading the middle-down method. So you can't use the quadratic formula. Yes You can always check it through a quadratic formula Then they will ask you categorically completing the completing the square method So then you know what to do you divide the entire equation by the coefficient of x square complete the square and then take under root and do not forget to take both plus and minus of You know the square root of the term which you will be obtaining So there is one mistake people generally do then third is a quadratic formula where It is always a good practice to write the value of a b c by the side of the paper So please do write what is a what is b what is c in in this in this what happens is let's say There is a negative number and they will try to trick you with giving some negative number and usually people You know forget that negative sign. So it's always good practice to write a what is a then b and then c then write the quadratic formula and then Then write the full, you know, uh, whenever you are deploying the number you should be careful for the sign So hence these are the three question three type of questions It could be asked direct questions solving differential equations. Sorry quadratic equations. Now second point is Questions on nature of roots. So second type of question will be Nature of roots. So, you know on nature of roots. So, you know, what is it? So you have to deal with discriminant Yeah, so discriminant is b square minus four ac So be careful. So they will be, you know, do not write b square plus four ac which I did in some time ago So you must be very very careful of the sign which you are using and you know, uh, remember it properly, so hence D equals to b square minus four ac and then either it is greater than equal to zero or less than zero basis that they will ask you Under what conditions the nature of roots would be equal unequal or unreal or not real imaginary all those things would be there they will For example, if I go back to the slide, let me see I have put one question, which was a one marker. Usually a one marker or two marker question is then always a World problem never mind. So we'll come to the world problems as well So, you know, so the one market question could be that in fact, I have shot a few videos If you go if you go to our, you know, youtube channel, you can see all those questions You know solved there so you can always go through the previous question Questions there. Yeah, so nature of roots third will be Third will be off word problems, right? So there could be word problems on On quadratic equations, right? So they will give you so let us take one word problem Which was asked in previous year and let us try and solve that question. So hence here is the question the question says The numerator of a fraction is three Less than the denominator. I have saw this question also and have posted the solution online YouTube channel so you can go there and see that so the new the numerator of a fraction is three Wait a minute. I'm sorry three less than Yeah, so the numerator Of a fraction is three less than the denominator Okay, the numerator of a fraction is three less than the denominator. So I will highlight this three less than the denominator So if I now know if x is the numerator The numerator of a fraction is three less than the denominator. That means denominator is x plus three Okay, this is the fraction If Two is added to both the numerator and the denominator. So let us add two So it hence is x plus two and the denominator becomes x plus five And this is the case two is added to both the numerator and the denominator and the sum of the new And the original fraction is 29 by 20. So hence, what can I say? Okay, so hence it is let me, you know, uh So just 29 by 20 is the sum. So let me go to this slide and then solve it here. So let us say So Yeah, so yeah, so we'll solve it here. Okay, so let us say Uh, they're saying that numerator is three less than the denominator and two is ordered two Both numerator and denominator and the sum of the two Is 29 by 20. Let us Please read the question at least two times to make sure that you do not misread the question The numerator of a fraction is three less than the denominator So x upon x plus three and two is added to both the numerator and the denominator Then the sum of the new fraction and the original fraction is 29 upon 20 And you have to find the original fraction very good. So basically now we have to find out x upon x plus three Clearly this is not a quadratic equation in this form But it can be reduced to one how let us do take the lcm So hence if you say take the lcm x plus three and x plus five on the denominator and here it is x times x plus five And here it is x plus two times x plus three And this is equal to 29 upon 20 guys 29 is a very hard No, it's a prime number and you know, uh, the moment you see number like 29. There's a Thought in the mind that it will be difficult to multiply with 29. So be very very cautious. So hence there So one thing which happens in these question is what is that and is you are prone to do careless mistakes or calculation mistakes Please always remind yourself that there is a possibility of careless mistakes. So be very very cautious Okay, so let us You know, uh, try and solve this So x times x is x squared plus five x then here If you know the identity you can use what is the identity? Let's say x plus a into x plus b I know the identity is nothing but x square plus a plus b x plus a b Right, if you know that then you can easily do do this multiplication or you can do a proper step by step multiplication So I will say if you don't remember then you multiply step by step. So anyway, it is x square plus three x plus two x plus six Right, and it is always a good practice to count the number of terms also so that you don't have you you shouldn't miss any particular term So, uh product of two factors which two terms each will give you four terms So see I have got four terms, right and in the denominator again, I should get four terms. What is it x square plus five x plus Three x plus 15. So four terms, right and then here it is 29 upon 20 Okay, now simplify x square plus x square is two x square And five x plus three x plus two x is plus 10 x Isn't it and then it is six And then in the denominator it is x squared plus eight x plus 15 equals 29 upon 20 now is the time To do cross multiplication. So hence it is 20 time two x square plus 10 x plus six. I hope I am not making any mistake On 29. Any ways we'll be checking. So 29 x square plus eight x plus 15 exactly This is this will be the scenario where you will be writing the exam So let us now calculate it is nothing but 40 x square then Then it is 200 x Isn't it 20 to 2 is 40 so 20 200 and then it is 120 And this is 29 x squared plus Eight into 29 is nothing but eight into 30 minus eight. So this is 232. I believe So eight nine is 72 seven daddy seven 16 plus seven 232 and there is an x And 29 into 15 is 30 into 15, which is 450 minus 15 that is 435 so 435 right Hence if you simplify 40 minus 29 is 11 x squared 200 x plus or minus 232 x is minus 32 x. I believe I hope I'm not. Yeah, and then 120 minus 435 is How much it is 235? Oh, sorry. This is 215 Right five and one two three and two Uh, sorry, 350 I know 120 minus 435 is So check multiple times so that no worries, you know, uh, because if you make one mistake and then the entire thing will go for a task Yeah, 315 equals to zero Okay, so this is my equation. Let me write the equation afresh Or let me get some space to write off Okay, so the equation is What is the equation equation is 11 x square minus 32 x minus 315 is equal to zero Now either we can go for a quadratic formula But then I am I say, you know, let us try and see if I can get up. So 11 into 350 This is a negative 11 into 350 this is ac right now. We have to split in such a way That I get a sum of 32 right right. How do we do it 11 is a prime number? So let's factorize 315, which is nothing but 3 into 105 or 9 into yeah, so it is nothing but 9 into 30 5 9 to 35 right so hence it is nothing but Minus 11 into 3 square into 7 into 5 Correct. This is mine. Yeah, so now I need to find out I need to break it in such a way now again If you see ac is negative So now if ac is negative I have to break this b in such a way that it is a Sum of two numbers, but one number has to be greater than the other in this case Okay, this is a this is the case right now. Let us see so 9 and 7 9 5 45 and 63 Okay, so if you see 77 and 9 half 77 and 45 I think Uh, so 77 and 45 Yeah, so if you see 77 minus 45 if you see 11 into 7 is 77 And 9 into 5 is 45 75 7 minus 5 is 2 7 so 32 right so it works So hence I can say it is 11 x square minus 77 x Plus 45 x Minus 315 equals zero Okay, so this means this means this is 11 x x minus 7 Plus 45 x minus 7 equals zero check with 7 into 45 is 3 and 15. Yes Uh 7 into 5 yeah 3 and 15 right it's correct. So hence it is x minus 7 and 11 x plus 45 equals zero All right, this implies either x equals to 7 or x equals to negative 45 by 11 Now clearly the fraction was You know the the term has to be positive right it was said that it is a integer Let's go back to the question once and see what was the question So the numerator was three less than the denominator if so hence it has to be you know, uh an integer So hence clearly I get one integer. So x equals to 7 is one solution. So hence the the fraction will be 7 upon 7 plus 3 that is 7 upon 10 now quickly check whether this is correct or not. So 7 by 10 plus Add 2 to numerator and denominator must be equal to 29 by 20. Let's check whether that is correct or not So 10 and 12 if you see 60 is the LCM So this is 6 into 7 42 and then this is 12 into 5 and 95 is the 45 So and it is 87 By 60 so divided by 3 you'll get 29 by 20. That means it is correct It's correct. So hence the fraction is 7 upon 10. This was a previous year question paper So these are usually the questions which are asked in you know quadratic equation Uh topic let us take a few more and see Whether we are okay Okay, so let me give you let me solve another problem, which is Yes Okay, so let me take another cvsc problem now the question is Let me go to a new slide Okay, so now the question is uh Find the values of k for which the given equation has real and equal roots and this is cvsc 2000 Cvsc 2015 question Even if the pattern has changed the nature of question will be similar So the question is Find the value of k Find the value of k k Such that such that Such that k plus 1 k plus 1 x squared minus 2 times k minus 1 k plus 1 x squared minus 2 times k minus 1 x Plus 1 equals 0 again read the question carefully again again Don't misread the question. So find the value of k Such that the equation k plus 1 x squared minus twice k minus 1 x plus 1 equals to 0 0 has has real Real And equal roots Equal roots now this is another way of testing, you know, what they will do is they will give you A quadratic equation within a quadratic equation in terms of you know So you uh in this question also though you have to find out the coefficients here But actually it will lead to let's say, you know use of what solving a quadratic equation itself So what is a condition for real and equal roots? You know that condition number two that means d must be equal to zero. What was d? D was b square minus 4 ac. So b square minus 4 ac Must be equal to zero. Yeah, so what is what did I tell you write a b and c? So what is a? a is k plus 1 right b is negative twice k minus 1 and c is clearly 1 c is clearly One right now, let us find out the value of d so b square so hence write 2k minus 1 and do not hesitate to put as many braces as possible. So twice square minus 4 into a a is k plus 1 and c is 1 so 1 and this must be equal to 0 so for real and equal roots, this must be satisfied now If you see what is it if you square it the negative sign has got no effect. So it is 4 k minus 1 whole squared minus 4 k minus 4 must be equal to 0 Is that it? I have expanded it and now let us expand this What is it? 4 times k square minus 2k plus 1 a plus b plus you know a plus b whole square Is a square plus twice a b plus b squared So hence you do this minus 4 k minus 4 equals 0 So if you if you simplify this, this is 4k square minus 8k Plus 4 right keep keep you know checking again immediately immediate check also helps a lot to eliminate any careless mistakes We quickly go back and see 4k square minus 8k plus 4 quickly and then minus 4k minus 4 Equals 0 now. What is it? This is 4k squared and then minus 8k minus 4k is minus 12k do not make mistakes in adding with signs And then 4 and minus 4 is 0. So it's you know Like this this makes our job a little easier because now you don't need to go for any You know a complex method of solving the equation why because the coefficient is 0 So hence what you can do is you can plug it plug out what 4k Is common. So hence it is k minus 3 equals 0 So hence you get either k must be equal to 0 Or k must be equal to 3 Okay, so either k must be equal to 0 Or k must be equal to 3. Let us check whether that is actually the value the case So if you if you put k equals to 0 in this equation, what will you get? You will get x square minus x square plus 2 x plus 1 equals to 0 which is a identity if you see If you put k equals to 0, this is a check right. We put k is equal to 0. You will get x square Plus twice of x plus 1 equals 0 which is nothing but x plus 1 whole square is equal to 0 that means it has Two roots both are equal. What are the roots x equals to 1 and 1 both are equal though Hence for k equals to 0 it is valid Let us check whether it is valid for k equals to 3 also. So in case x k equals to 3 Then what will happen? It will be 4 x square minus 4 x Plus 1 indeed. It is 2 x minus 1 whole square is equal to 0 So x equals to half and x equals to half are the two equal and real roots. So hence Hence both the solution that is k equals to 0 and k equals to 3 Will satisfy the condition and the equation will then have Real and equal Roots. Okay, so let us take another problem from previous year paper So usually these will be let's say, you know, either at one marker or a two marker question mostly Okay, now the question is there's a interesting question And though it is not a previous year question, but then it could be asked in exams. So let us say The question says Yeah Show that the equation the question is Show that Show that The equation Show that the equation x squared plus A x x squared plus A x minus 4 equals to 0 has real has real has real and distinct distinct roots Distinct roots Distinct has real and distinct roots for all real values of a for all for all real values of values of A again read the question once again make it a habit show that the equation x square plus 8 x minus 4 Sorry, not 8 x x square plus A x minus 4 equals 0 Has real and distinct roots for all real values of a okay, so clear So what is the condition for real and distinct roots? You know d must be greater than 0 right not even equal it has to be greater than 0 Okay, so let us find out d d is nothing but for a quadratic equation. It is b square minus sorry 4 ac Is it now? What is b here? The b value is a if you see don't get confused here the b value is a so hence I will write a square minus 4 ac so minus 4 times 1 a is 1 in this question. So this don't get confused by this a Yeah, this a is nothing but b in our general equation and here the a value is 1 and c value is negative 4 So hence negative 4 Is equal to a square plus 16 Correct now a is a real number A is a real number is it a so hence a square is always greater than equal to 0 Because real number squares cannot be negative right so a square is always greater than equal to 0 So clearly a square plus 16 is always greater than 0 Now it cannot be said as equal to 0 because 16 is added to a non-negative number So a square plus 16 is always greater than 0 whatever is the value of a isn't it so hence d is always greater than 0 hence it is a condition for e are real sorry it is a condition for real and distinct distinct roots Fair enough. So this is on nature of roots. Let us now take few more different type of question Okay Another question. Let's say let's take another previous year question. So the latest or other, you know, so let's say take let's take a cvc 2014 Okay, now the question again is on the nature of roots Nature of roots and the question is asking this Uh find the value of k so again find Find the value of k Find the value of k for which for which for which for which the roots for which the roots roots are Roots are real and equal real and equal for the following for the following equation Okay, and what is the equation given in cvc 2014 this question was asked and this is px times x minus 3 Plus nine equals zero. This is which year this is cbc 2014 not that you know, uh Not long back now again read the question once more The question says find the values of k for which the roots are real and equal again real and equal For e for following equation. What is the equation px times x minus 3 Plus nine equals zero now if you see there is no Direct x square term which you can see but it is it is right So, you know, uh, don't get you know confused You can see that if you expand this you will get a square term So if you see it is nothing but px squared Minus thrice px Three times px plus nine Equals zero now for real and equal. What do I know d must be equal to zero for real and equal Roots Is it it so that is so then what is d d is b square minus 4 ac must be equal to zero Now what is b b in this case is negative 3 p if you see negative 3 p is b So b square that is square of this minus 4 into p into 9 must be equal to zero So if you see this is 9 p square minus 36 p equals zero You can uh, you need not multiply it here also because there is a you know, you can extract a common factor So you can now say 9 p into p minus 4 Isn't it equals zero So if this is the case then either p equals to zero Or p must be equal to four again, there are Two solutions to this. So let us check if p equals to zero Actually p cannot be zero if you see p cannot be zero why in a quadratic equation the The coefficient cannot be zero. So hence you have to write p cannot be zero or you can write p can't be zero Can't be zero Can't be zero because you name it as one and say because because One is a quadratic Equation But hence let us check. What is whatever if p equals to four. So if p equals to four the equation becomes 4x square Let us check So let us check 4x square Minus this check you don't do while solving the question in the main sheet. You can always do it in the rough So 4x square minus 3 into p p is 4 again into x plus 9 isn't it? So which is nothing but this is uh Yeah, so this is 2x 4 square Minus Yeah, so this is 2 into 2x into 3 if you see yeah 4x into 3 is this plus 3 square This is equal to 0. This is equal to 0 and hence it is 2x Minus 3 whole square is equal to 0 So hence there is only one solution Equal roots right. What is that x equals to 3 by 2 so hence p is equal to 4 Is the right? Okay guys, so we saw this question also needs on nature of roots. Let us go and solve some word Problems now. Okay. There are different types of word problems. We'll take up one by one each, you know, uh We just uh saw numerator and denominator kind of a Yeah, uh, yes, let us take up some previous year questions First is on digits of one. Yeah, so let us take up some previous year questions Next is now meanwhile guys if you have any question in uh in your in your Mind or let's say if you are not able to solve anything you can always post it here Or through whatsapp and uh, we will be able to solve it and post it here Okay Let us take another question. So question it was asked this question was asked in 2006 Question was asked in 2006 CBC. What is the question question is a two digit number a two digit number Two digit number is such that is such that The product of its digit That the product Product of its digits digits is 18 Okay, when 63 When 63 is Subtracted when 63 is subtracted from the number From the number From the number the digits interchange their places The digits Interchange Their places Their places so you have to find the number Find the number once again read the question once again so that you don't make a mistake Okay, the question says a two digit number is such that the product of its digits is 18 When 63 is subtracted from the number the digits interchange their places find the number Now this is interesting interchanging of the digits, right? So let's say two digit number is such that the product of its digit is 18 So if one digit let you write like this let the Let one of the digits One of the digits Be x Then the Other digit Will be 18 upon x Now in such questions, you know that digits will be an integer right and digit what integer from 0 to 9 but clearly x is not 0 by because if x were 0 I'm sorry guys. So if x is 0 then You will not get the product of the two digits as 18 Also x cannot be 18. Why? We can do is this thing should go in your mind parallely You don't need to stop and think like this, but I'm just giving you a hint to How to eliminate your you know or how to get to answers And you can it is also can help you in let's say checking whether you have done it correctly or not six clearly cannot be 18 because if you see 18 is the product of two digits two digits 18 so 18 can be one into 18 18 can be equal to two into nine. So here is one possibility 18 can be also written as three into six another possibility Right And then six into three and then nine into two and then 18 into one again. So hence if you see either it will be You know so either of these solutions will be right Okay, but let's Let's go ahead and See whether we are doing it correctly or not so Yeah, now the second is When 63 subtracted from the number. So what is the number guys? Let's say x is the unit space Okay, so let's say x is in the unit space or one of the digits bx. Let's say you also write unit space Unit space Okay, so what is the number? So the number is The number will be nothing but 10 into 18 upon x plus x This is the number right why because if you if you are having a two digit number Let's say ab then the value of the number is a into 10 plus b all of you No, this i'm not writing i'm explaining it again. So this is the number 10 into 18 upon x plus x now What are they saying if 63 subtracted from the number the digit interchanges? So let us subtract some 63 from it. So 180 by x Plus x minus 63. So we can do that that the number the digits interchanges space So units becomes 10s and 10s becomes units. So it will be now 10x plus 18 by x So this is the question now once you are done with Formulating an equation your job is half done. You are now You have to just solve the equation. So let us see try and solve Okay, so once the you know, it is a better practice to check once again whether the Thus the formulation of the equation is correct or not. So if you see this is 10x So 10 times the digit at the 10s place plus the units place digit minus 63 is equal to the reversal of digits Okay, looks good. So let us now solve it by Collating all the x terms together. So if you see this is nothing but 180 by x And so let's take everything on the right hand side. This means that 10x minus x is 9x Okay, so this x will go on to the Right inside if you are not comfortable with that then you Please write full step full step will be this at 10x plus 18 by x And then this becomes minus 180 by x this becomes minus x and plus 63 was zero Okay, but directly also you can write as what so 9x and then 9 and then 180 minus 18. So this is nothing but 162 by x right and then plus 63 equals zero Is it it so 9x minus 162 by x plus 63 equals to zero you can make it it is not a quadratic form right now But you can reduce it to one. So it is 9x squared minus 162 plus 63 x equals zero Right, so hence it is x 9x squared Yeah, so if you see you can actually Take out one nine from everywhere Isn't it all are multiples of nine all the course is the multiple of nine So you can say x square strike of nine minus this is 18x for 18 and this is 7x Equals zero. Yeah, so hence it is x square plus 7x minus 18 equals zero This implies you can write x square minus 9x Plus 2x sorry the other way around So hence this is what I was saying Please be very very every step you just do a mental recalculation. So plus nine and minus 2x minus 18 equals zero This implies you can take x as common from the first two terms. So x plus nine And this is negative two common from the second last two terms x plus nine equals zero and hence I'm writing it here now. It is nothing but x plus Nine times x minus two equals zero x equals to negative nine or x equals Two right clearly x equals to negative nine is not a solution because we are talking about digits digits are between zero to Nine both inclusive so x equals to two will be the right answer. So when x equals to two The number is 18 by two nine three two. So hence the number is 92 Right 92 now if you subtract 63 from here What will you get you will get Now 29 you should get 29. So if you see 92 minus 63 Is actually 29. So hence the number the digits are interchanging its place, right? So hence our solution is correct Okay, so this was another type of problem. Let's go to another one Another different type. Okay So let us solve another. Yeah now Quantity equations for solving problems on time and distance will take up One time and distance problem. Also, this is also very common commonly asked. Okay Yeah So let me use the space. Yeah, so the question again question is This and I'll take up a Previous year paper question. So this is again 2006 paper which is there. So there is Absolutely no problem in solving a little older questions also Why because the pattern of the question remains the same or rather we can take up A recent one. So 2000 the question is a motor boat a motor boat motor boat who speed in still water To speed in still water still water Is is 18 kilometers per hour also be mindful of the units. Okay Takes takes one hour more One hour more To go to go 24 kilometer 24 kilometer upstream upstream upstream then then To return down return downstream downstream To the same spot To the same spot again Find the speed of the stream fine. You have to find the speed of the stream Okay, so once again a motor boat whose speed in still water is 18 kph 18 kilometers per hour It takes one hour more to go 24 kilometer upstream that to return downstream to the same spot Find the speed of the stream Okay, so will so whatever is required you assume that to be the variable that is the common Trend or common practice find the speed of the stream. So I'll say let the Let the speed speed of the stream Be x kilometers per hour write the units here itself so that even if you'll miss writing units little you know in the last step Then you know, it should not make or you should not attract any analogy now A motor boat whose speed in still water is 18 kilometers per hour obviously when it is Going downstream its speed will be more why because the velocity of the water will aid the velocity of The motor boat and while it is going upstream it it will have a Less speed is it in so now it is saying one hour more So what is what is you know, what is constant here the constant is the speed sorry the distance, right? So this is traveled upstream is equal to distance traveled downstream. So what is distance traveled? distance Oh, okay, 24 kilometers given actually anyways one hour more to go 24 kilometer upstream Okay, so 20 distance is 24 kilometers So distance is equal to speed into time. We know that in case the If speed is constant then distance is speed into time. So hence 24 kilometers is equal to speed of Let's say while going upstream. Uh, it takes time p Okay, so so the speed was x and It is a drive. So hence Huh, so what will happen? So let us write this thing separately. So hence velocity upstream that means when the Or the speed upstream and the steamer or the boat is going upstream the velocity will be Um 18 minus x kilometers per hour, isn't it? kilometers per hour why Because the velocity will be reduced because the motor boat has to go against the speed of the stream Okay, now And velocity downstream Downstream is equal to 18 plus x Kilometers per hour Isn't it 18 plus x kilometers per hour now So while it is going upstream. So distance is 24 into Um Yeah, so time below so hence we will say time upstream Okay, time upstream is equal to nothing but Distance by speed isn't it? So distance is 24 and speed for upstream is 18 minus x Okay time downstream is equal to 24 upon 18 plus x now if you see In both the t up and t down 24 the numerator remains same denominator is more in the second case. So hence t down is less which is quite obvious The denominator is more here denominator is less here. So hence t down that means going downstream the speed Sorry time taken will be lesser now. There's a difference between the two times Obviously this is more isn't it t up will be more than t down But there's a relationship given what is that it takes one hour more to go up That means I can write T up is equal to t down plus one Isn't it so hence what is t up? So let's now form the equation. So 24 upon 18 minus x Is equal to 24 upon 18 plus x Plus one so time I calculated time up I calculate time down and then I know time up is one hour more To go upstream, right? So this is my so you will get 50 percent of the marks here itself when you are writing the equation correctly Now the the question remains to solve this the other half is to solve the question correctly now. So hence let us solve this Now how to solve this you have to write 24 upon 18 minus x minus 24 upon 18 plus x equals one after rearranging Take lcm. So it is and 24 is common. So you can write 24 into 18 plus x And then here it is and in the denominator first write the denominator. So 18 minus x times 18 plus x denominator and hence 24 is common so 18 plus x minus minus 18 Minus x like that, right equals One now. What is it? So if you see 18 minus 18 will get cancelled. It will become 2x In the denominator. So 2x so hence it is 24 into 2x upon 18 minus x times 18 Plus x equals one Okay, so hence if you see If you see the next is Solve this this is 48x is equal to 18 minus x and 18 plus x Okay, so this is 48x Is equal to 18 into 18. So I'm not multiplying. I'm just keeping it like that because you know, we'll see if we can factorize it later on And then oh, so it is 18 square minus x square Then and simple 18x minus 18 plus x is a square minus b square. Now the final equation is x square minus. Sorry plus 48x Minus 18 into 18. So I am purposefully keeping it. I'm not multiplying because anyways, you have to factor it Okay, so let us see How do I split 48 to get 18 into 18? Okay, so hence what is 18 into 18 18 into 18 is 81 into 4. So if you see this is nothing but 18 into 18, what is it? It is nothing but 3 square into 2 Into 3 square into 2, right? So now you know 81 into 4 that which is nothing but 4 324 Yeah, but You have to break this 18 into 18 in such a way that you get 48 So obviously since it is a negative number So you have to break in such a manner that it is 48 is difference of two numbers whose product is 18 into 18 Okay, so let us try and see if we can get 48 here So if you see it is um How do we so let's say 9 into 4 So Yes, um 6 So 9 into 4 is what is yeah No, um So how do we break it? So if you see um 40, right? So hence I have to let's let's break them. So hence clearly 3 I am doing it here. So 18 into 18 can be written as 3 into or let's start with 2 so 2 into what is left 9 81 into 9 9 9 into 9 81 81 into 2 is 162 this this will not work then 3 into this is also not is not going to work. Why because it is 2 uh 2 2 2 bigger number for having a difference of 48. So hence we have to find let's say 3 into 4 Is 12 and here it is No, we'll have to 6 into 6 and rest is 54 and 6 is the number if you see how 3 into 2 is 6. So 6 if you see 6 and 54 Is the other factor right so 6 into 4 is uh 6 into 54 is 3 24 Correct. So hence this will work and 54 minus 6 is 48. So hence I get x square plus 54 x minus 6 x minus 3 24 equals 0. So this implies this implies x into x plus 54 minus 6 times x plus 54 equals 0 and hence Yeah, so hence if you see this is nothing but x plus 54 and x minus 6 equals 0. So either x equals to negative 54 or x equals to 6 but speed cannot be speed cannot be negative right this cannot be negative. So hence we say speed cannot be negative Can't be negative So hence Solution is x equals to 6 kilometers per hour, right? This is the solution. This was asked in 2014 CBC let's take up another type now it is nothing but On ages let's say there's a question on uh age, okay, so this is another type you'll get Let us solve this. This is again a CBC 2010 question CBC 2010 question. Yeah question says A girl is twice old as her sister a girl is twice as old as her sister Yeah, you have to convert these into expressions right mathematical expressions four year four years hence four years Hence now in in four years hence the product of their ages the product product of their ages The product of their ages will be it's mentioned in years will be 160 160 Find their present ages find their present present ages Present ages Present ages So a girl is twice as old as her sister. So hence you have to be careful about you know Years hence Many of you what you do is you augment Now so So we're saying The ages Of the sisters be x x and x and x plus So twice x right so x includes So go is twice as old twice as old as her sister right x into its four year hence The product of their ages will be 160 so four year hence The ages of sisters Would be how much would be x plus four and x plus four another product that's saying x plus four and two x plus four is equal to 160 So you have to find out What is the value of x so let us simplify this this is two x squared plus four x to expand plus four x plus eight x Plus eight x Plus eight x and then 16 equals one 16 Then quickly check whether it is correct or not. So what is it? Two x square That's correct then four x that's correct four into two eight x is that's also correct and four into four is 16 Which is equal to 160. So this implies how much two x squared plus 12 x Plus or sorry minus one 44 Equals zero you can eliminate two from here You can eliminate two from here. This is the question This is the quadratic equation. So half the marks will be given here itself if you have done it correctly So hence eliminating two you'll write x square plus six x minus 72 equals zero Yeah, you can write dividing dividing the equation by two Isn't it now it boils down to solve this question. So hence it is if you see 72 So 72 is to be broken in such a way that Such that the product is 72 but in the sum is 6. So there's no brainer here You can do it as plus or plus 12 x minus 6x minus 72 is equal to 6 zero why because 12 6s are 72 if you now do this take common. This is x plus 12 minus 6 x plus 12 equals zero So this is x plus 12 into x minus 6 equals zero so hence So hence my dear friends x is equal to negative 12 or x equals to 6 But clearly this is not a feasible solution not a Feasible Solution why because age cannot be Age cannot be negative So x equals to 6 is the right solution Okay, so the ages of the sisters will be ages of the or you should write present ages present ages of the sisters are 6 and twice of 6 2 into 6 equals to 12 years This is about You know age is problem. Okay Now there will be few application of the same in geometry, you know, you can use Pythagoras theorem There's illustrations in that then in mention they can also combine A problem in geometry or menstruation. So we'll take up one of that sort So let us see what is other type of other application. So one is uh, you know again cbsc 2000 cbsc 2015 question Now this question says the area the area of An isosceles triangle isosceles Triangle is 60 centimeters square and and the length And the length of Of each one of its Of its equal equal size Equal size is 13 centimeter Okay, find its base Find its base Let's read the question once again the area of an isosceles triangle is 60 centimeters square And the length of each one of its equal size Is 13 find its find its space very good. Let's try and solve this problem So isosceles triangle is Let's try and solve this problem So isosceles triangle we'll have to make. Okay, so this is my isosceles triangle. That's it So in geometry problems, please do not forget to make a drawing map. Okay now The area of an isosceles like so what is an area of an isosceles triangle? So you'll have to drop a particular from here So always draw always draw a figure whenever you are solving a Uh Geometry problem. Okay So this is a rough diagram where you do, you know, you can use it for This purpose now What is it given? Right. So each of the equal size. So this is 13 centimeter 13. So let's say a b See, so it's given. What is given given is A b is equal to ac is equal to 13 centimeter Okay, and area of triangle a b c Is equal to 60 cm square Okay, now I have to find to find what To find what b c Okay, now construction. What did you do construction drop? Are you right here a d? ad perpendicular to b c Okay, now clearly in an isosceles triangle, you know that bd is equal to bd is equal to dc, right? You can write the reason in an isosceles triangle Triangle the perpendicular from From the vertex vertex bisects the Non-equal side Okay, so so let us say bd is equal to right. So let us say bd let bd is equal to x Okay, so to find bd now if bd is equal to x then ad Is equal to nothing but under root 13 square minus x square Is it it ad is 30 by and you write the reason by Pythagoras by Pythagoras here Okay Now what do we now Do so I now know the edge so area area of triangle. So half into 2x half into base into height That is ad will be equal to 60 that is what is given Okay, so hence half into 2x. So I can just cancel this 2 of so x into ad what is ad under root Is equal to 60 Isn't it so hence if you so and then you'll write What will you write you will write squaring Both sides these steps, please do not forget squaring both sides Right you'll get x square into 13 square minus x square is equal to 60 Is it it so hence it is nothing but 13 x square minus x to the power 4 is equal to 60 Or you can write x to the power 4 minus 13 x square Is equal to or sorry plus 16 Is equal to zero Right, so I hope we are doing the 13 square minus x square Yeah, half into 2x Half into oh wait a minute. Yeah, half into 2x Into ad which is 13 square minus x square is 60 so hence hence you will get x to the power 4 minus 13 x square Oh, no, uh, wait a minute square. This will be this is a mistake. This is square So this will not be 60. This will be sorry. This will be 3600 Plus 3600 Equals to zero so please be very very Careful. Oh, this is also 13 square right. So please be very very careful when you are Doing this so hence it is nothing but x to the power 4 minus 169 x square Plus 3600 equals zero Okay, so again either you can go for uh quadratic formula thing and also it is a bico quadratic equation. So you can say let x square be y you can solve it here itself or you can simplify by saying let x square equals to y So the equation is reduced to y square minus 169 y plus 3600 Okay, so now if you see if you if you try to split 3600 so 36 into 100 is one But 100 plus 36 is not 136. So we'll have to do a little bit more. So if you see it is uh Also This is actually 44 into 125 Yes Yes, so 125 into 44 actually will give you Uh 3600 Yes, that's fine I think it is correct I'm sorry not 45 no no no sorry the other way around 144 into 25 125 doesn't divide 3600. So it is 144 into 25. Yeah Yeah, that makes sense. So 144 into 25. Yeah, is it that is yeah. So this is equal to zero So hence Hence what will you say? You'll say y square minus 44 y minus one sorry 144 y minus 25 minus 25 write it properly don't overwrite so yeah minus minus 25 y plus 3600 equals 0 okay yes so hence it is y times so I am writing it here y times y plus 144 so y times y plus 144 and then y plus 144 anyways will be taken out and here it will be simply minus 25 equals to 0 am I right not oh sorry wait a minute there is this thing yeah so that's what the problem is please write it very clearly so it will not be plus it will be minus right and here also this is minus so hence it is y minus 144 into y minus 25 equals 0 so either y is 144 or y is 25 these are the two possible solutions so if y is 244 guys then what will be x x square was equal to y isn't it so hence we can write that means x square is 144 so x equals 12 I am neglecting minus 12 why because negative 12 cannot be dimension of a triangle so hence x equals 12 and from here x is square equals 25 so x will be simply 5 again I am ignoring the negative values ignoring x equals to negative 12 these all these will definitely be the solution to these equations but they will not be feasible solution x equals to negative 5 ignoring these right so x x is 5 or x equals to 12 centimeter so hence hence you have to find out the base so for the lack of space I am writing it here so the base BC will be twice of x that is either 10 centimeter or 24 centimeters okay this is how you will be solving these sums so I thought I think we have covered a lot of problems today we also covered the theory part so I think you guys should be good now and as I told you there will be typically three types of problems which will be asked one is solving the equation by any of the methods second will be on the nature of roots and third will be word problems where again solving of the equation will be important as you know very important ingredient of the problem so if you are you know solving all the previous year questions around 20 odd questions you know before you take the exam I think you should be good and in case you have any difficulty any doubt or any concept has to be revisited please reach out to me and if needed we will be doing another dedicated session on the same topic I hope these revision processes are helpful to you so please attend these if you are not getting time or because of let's say your other mock exam going on school if you are not able to find time I would recommend that whenever you get time you please go through the youtube channel and see it watch it at your own pace so we will call it a day and I hope this session was useful to you and we will be back again with another class tomorrow so thanks for attending the session guys all the best