 In this video let's try to summarize the unit quadratic equations for class 10th. So we'll begin our journey by developing a basic understanding of what do we actually mean by these equations. And once we know what they are then we'll talk in terms of finding the solutions for these equations. And since we are talking about finding the solutions we'll see what do we mean by roots or zeros of a quadratic. And once we know what do we mean by roots or zeros then we'll see what are the different methods to solve these quadratic equations. In other words what are the different methods to find these roots or zeros. So we'll focus on three different methods factorization, completing square and the famous quadratic formula. And even out of these three we'll focus mostly on factorization and the quadratic formula because every time whenever you want to solve a quadratic equation we'll almost every time use the factorization method or the quadratic formula. And once we know how these methods work let's talk a little bit more about the roots or zeros. Basically how can we find their nature. How can we say whether the roots are real or unreal even without finding them. And once we have finally talked about the nature of roots then we'll spend a little bit of time on understanding the word problems related to quadratic equations. So this is how the entire journey would look like. And let's start now. So here you can see all the topics that we would be discussing in this video and you can also see their timelines. So in case you are clear with any of the topics for example let's say that you already know what do we mean by quadratic polynomials and quadratic equations. So you can directly jump to roots and zeros at the mentioned timeline. So let's start with our first topic of what do we mean by a quadratic polynomial and a quadratic equation. So a polynomial of degree two would be a quadratic polynomial. For example let's say four x square plus six x plus one. The degree of this polynomial is two hence this would be a quadratic polynomial. Let's take another example let's say that we have negative five m square plus nine. For this polynomial the degree is still two so this would be a quadratic polynomial. Even monomials can be quadratic for example negative of four by three y square. For this monomial the degree is two hence this would be quadratic. Now when you equate a quadratic polynomial to another quadratic polynomial or maybe a constant value or maybe a linear polynomial then you get a quadratic equation. For example let's say that we have seven y square plus two y plus eight equal to zero. So this would be a quadratic equation because even for this equation the degree is still two. Let's take another example for example let's say that we have 32x square equal to nine. This time we equated this quadratic polynomial to a constant value. Hence the degree of this equation is still two so this would be a quadratic equation. So now once we know what they are let's see what do we mean by standard form of a quadratic equation. So when you write all the terms of the quadratic polynomial on one side and equate that to zero and the terms are written in such a fashion that the powers of x are in decreasing order and here you can see that a is the coefficient of x square, b is the coefficient of x and c is the constant value the term without any x. So whenever you write a quadratic equation in this form it is said to be written in the standard form and also make sure that your coefficient of x square that is a is not equal to zero because the moment a would be equal to zero this equation would turn into a linear equation which we do not want. Now let's move on to the second topic of what do we mean by roots or zeros of a quadratic. The value of x for which the value of the quadratic polynomial is equal to zero is called the zero of the quadratic polynomial. Let me explain what this means let's take an example let's say that we have x square plus 2x plus 1. Now what value of x should I substitute here such that the value of this entire polynomial is equal to zero. Now it is given x equals to negative one would satisfy this condition and hence x equals to negative one would be the zero of this quadratic polynomial. Now what's the difference between zero and root? If we have a quadratic polynomial we call it a zero and if we have a quadratic equation we call it a root. Basically both mean the same thing. In other words for a quadratic equation you can also think of roots as the solutions for this equation. What all are the values of x that would satisfy this given equation? Those values of x would be known as the roots of this given quadratic equation. Now once we know what roots are let's see how can we actually find the roots? How can we actually solve this quadratic equation? Let's start with factorization method to solve a given quadratic equation. In other words let's see how we can factorize a given quadratic equation to get to the roots of the equation. So whenever you need to factorize always look out for these four things. These steps here would help you in factorizing given quadratic equation. Look for any common factors in the terms of the equation. Sometimes rearranging the terms can also help in factorization. Then we have splitting the middle term and we'll see how do we do this and sometimes using identities can also help in factorization. Let's look at them one by one to have a better clarity of what we are trying to do. So first of all we would like to start with pulling out any common factor from the given terms of the quadratic equation. For example in this case we can see that a is the common factor in all these three terms. So what we can do in order to simplify this equation further is that we can pull out one a from all these three terms. So what we would have here is a times a square x square plus 2x plus of 1 equal to 0. Now we can further think about factorizing this simplified quadratic equation but pulling out a made this clearer to factorize this further. Sometimes rearranging the terms can also help in factorization. For example in this case let's see what do we have here. We can write this as x square plus 2a plus of 2 times x plus of a times x is equal to 0. Now we can say that first two terms from the first two terms there is no common factor that we can pull out. In the last two terms we had x is the common factor but that would not lead us to factorizing this given polynomial. So what we can do let's do one thing. Let's try to combine first term and the third term and pull out any common factor and similarly let's try to combine second term with a fourth term and try to pull out any common factor. So we can write this as x square plus 2x plus of 2a plus ax equal to 0. From the first two terms here we can pull out x as the common factor so this would give us x times x plus 2 and from the last two terms we can pull out a as the common factor. So this would again give us x plus 2 equal to 0. Now we can see that we have x plus 2 as the common factor so we can pull that out. So this would finally give us x plus 2 times x plus a equal to 0 which is the factorized form of this given quadratic equation. Now let's talk about splitting the middle term. So this is one of the most common and heavily used methods for factorization. What do we do in this? Basically we try to find two numbers such that their product is equal to this constant value the term without x. The product should be equal to 12 and their sum should be equal to coefficient of x that is negative 7. Now the key to find these numbers is to think in terms of the factors of 12. For example here if we take 3 and 4 we know that their product would be equal to 12 but 3 plus 4 would give us 7 but what we need here is negative 7. So instead of 3 and 4 we can take negative 3 and negative 4 because negative 3 times negative 4 is still equal to 12 and negative 3 plus negative 4 would give us a negative 7 as our answer. So now once we have these two values we'll try to split the middle term in terms of negative 3 and negative 4 that is we can write this middle term as x square minus 3x minus 4x plus of 12 equal to 0 and now from the first two terms we'll try to pull out any common factor. So here we can see we have x as the common factor so this would give us x times x minus 3 and from the last two terms we can see that we have negative 4 as the common factor so we can pull that out. So we can have same x minus 3 as the common factor again so negative 4 when pulled out from last two terms would give us x minus 3 equal to 0. Now since x minus 3 is the common factor we can pull that out. So this would finally give us x minus 3 times x minus 4 equal to 0 which is again the factorized form of this given quadratic equation and from here we can easily find the roots because if this thing is equal to 0 either x minus 3 is equal to 0 or x minus 4 is equal to 0 so from here we can say that x is either equal to 3 or 4. Sometimes even basic identities could help in factorization for example in this case if we look at the first three terms carefully it is of the form a plus b whole square. Now how's that so let's see so on expanding a plus b whole square we would get a square plus 2ab plus b square. Now if a square x square is our first term we can write this as ax whole square and if this is our b square we can write this as 4 whole square and now we need 2ab which would be equal to 2 times ax times 4 which is actually equal to 8ax so first three terms we can write it as ax plus of 4 whole square. So we used this identity a plus b whole square to factorize the first three terms of this expression and now we have minus of b square equal to 0 now this is again of the form of difference of two squares that is something like something like a square minus b square which is equal to a plus b times a minus b so this we can further factorize as ax plus 4 ax plus 4 plus of b plus of b times ax plus 4 ax plus 4 minus of b minus of b equal to 0. So now we have factorized this given quadratic equation and from here we can find the roots and the values of x that would satisfy this. This is how we try to do factorization by following the four key steps that we just discussed. Now let's quickly see how we can use completing square method to find the roots of a quadratic equation. Since we don't use this method very often to find the roots we mostly use factorization and quadratic formula. I'll just leave an example here for you to understand how this method works and if you want to know more about this method you can definitely go ahead and watch the lesson based on completing square method in the unit quadratic equations. As of now let's move on to quadratic formula. So quadratic formula gives us a direct expression to find the roots of a standard quadratic equation. So it's important to change the equation in standard form to apply quadratic formula to get the roots. For example let's say that the equation is ax square plus bx plus c equal to 0 as per quadratic formula its roots would be given as negative b plus of square root of b square minus 4ac divided by 2a and negative b minus of square root of b square minus 4ac divided by 2a. The quantity inside the square root that is b square minus 4ac excluding the square root the quantity inside this is also known as discriminant and we later see how we can find the nature of roots based on this discriminant value. As of now this is the quadratic formula and I'm not a big fan to suggest you to remember such formulas but this is kind of really important because many times when you are not able to factorize the given quadratic equation or sometimes when you feel the need to check whether your roots are correct or not you can directly use this quadratic formula to find the roots. The only thing that you need to worry before applying a quadratic formula is that change the equation in standard form and do the correct substitutions. Now let's see an example based on quadratic formula. So let's say that we have this quadratic equation in standard form 1x square minus of 7x plus of 12 equal to 0. This coefficient of x square would be our a coefficient of x negative 7 would be b and the constant value here positive 12 would act as c. On doing the correct substitution of a b and c in the formula negative of b that is negative of negative 7 plus or minus square root of b square that is negative 7 square minus of 4 times a that is 1 times c that is positive 12 divided by 2 times a and a in this case would again be 1 2 times 1 would give us the values of roots that is 4 and 3. So this may involve heavy calculations at times but you would finally get your roots. You can also use quadratic formula to directly find the sum and product of roots. So basically when you will add both these values the final value that you would get is negative b divided by a that is negative of coefficient of x divided by coefficient of x square and this would directly give you the sum of roots and similarly when you'll multiply them the value that you would get is c by a the constant term here divided by coefficient of x square. So c by a would directly give you the product of roots without even calculating them. So this is helpful for questions in which they directly ask you to find sum and product of roots. For example in the previous slide we saw the roots of the quadratic equation x square minus 7x plus 12 equal to 0 equal to 0 and we saw that the roots are 3 and 4. So what would be the sum of roots in this case? Negative b by a that is negative of coefficient of x that is 7 divided by coefficient of x square which is 1 in this case so 7 by 1 which is indeed true 3 plus 4 is 7 and what would be the product of roots c by a that is 12 divided by 1 which is again equal to 3 times 4 that is 12. By now we have seen how to solve a quadratic equation using these three methods how we can basically find the roots using the methods given here. Now let's try to focus on the nature of the roots. Let's see without even finding the roots how can we say whether the roots would be real or unreal. So this is the value of the roots that the quadratic formula gives us for the standard quadratic equation. This can be real if and only if the quantity inside the square root should be real and how can that be possible? If the quantity inside the square root is non-negative then this entire root would be real and if the quantity inside this root is negative then we'll see that the value of these roots would be unreal. So the major deciding factor for the roots to be real and unreal is the quantity inside the square root that is the discriminant. So if the value inside the square root b square minus 4ac is greater than 0 then the roots are perfect then they are real and they would be different we would get two different real values of the roots. If this quantity inside the square root is equal to 0 then we would still get real roots but they would be equal and the roots would be negative b divided by 2a. The only problem is if the quantity inside the square root is negative in that case the roots would not be real and hence we would get unreal roots. So without even actually calculating what the roots would be we can directly calculate the discriminant value to find whether the roots would be real or unreal. Let's look at some examples here. For the first equation here let's try to find the values of a, b and c. a is the coefficient of x square, b is the coefficient of x and c is the constant value. So for the first equation b square minus 4ac that is the discriminant would be equal to 3 square, 3 square minus 4 times a that is 4 again times c that is 2. So 9 minus of 32 less than 0 hence the roots would not be real. The roots would be unreal in this case. Let's take the second example. In this case b square would be 2 square minus 4 times a that is 1 times c that is negative 8, negative 8. So this would give us 4 plus of 32 which is greater than 0. Hence this is greater than 0. We would say that the roots are real and unequal. And last case here b square minus 4ac would be negative 2 whole square minus 4 times a that is 1 times c which is again 1. So 4 minus 4 is equal to 0 which is the second case. Roots would be real but equal. So we are done with the chapter now. We now have all the tools that we need to solve any equation based on quadratic equations. This part of the video is mostly dedicated towards how you can approach a word problem based on quadratic equation. So basically when you see a word problem always try to think in terms of these three steps. You need to use the given scenario to get the equation. You cannot directly see the equation given. You always need to use the given scenario given English statements to get to the mathematical equation. Give some name to the unknown and reread the question. So wherever there is the unknown whatever they are trying to tell you to find give it a name call it x call it y and then try to reread the question in terms of the variables x and y. And finally try to form equations using the variables and the relation given in the question. Let's look at some examples here. A two digit positive number is such that the product of its digits is 6. If 9 is added to the number the digits interchange their places. Find the number. So this is the unknown. We need to find the number. So how can we assume a two digit a two digit number? So basically we are talking about the digits at units and tens places. So let's say that the digits at tens places x and the digit at ones places y. So what would the number be? 10 times x number multiplied by its place value plus of one times y. So this would be our original number. Now let's reread the question. The product of its digits is 6 that is x and y. The product of x and y is 6. If 9 is added to the number the digits interchange their places. So 9 is added to this number that is 9 plus of 10 x plus y. What happened now? The digits interchanged their places. So instead of x we had y and instead of y we had x. So what would be the number look in this case? Digit multiplied by its place value that is 10 times y plus of one times x. So this would be the interchanged form of the initial number. So when 9 is added to the initial number this is what we get 10 y plus x. So we have two equations. From here we can write that y is 6 upon x and this value we can substitute in the second equation and finally simplify this to get the values of x and y. So this is how you should try reading the question. Try to assume the variables and try to reread the question until and unless you get to the equation and once you have the equation then it's mostly simplification. So yeah that's it we are done. We have revised the entire chapter quadratic equations and I think now you would be much clear and much comfortable with all the topics that we discussed in the video.