 Hello friends, so continuing with this session on quadratic equation, now we are going to discuss how to solve a quadratic equation by completing the square method. So this particular expression is going to be occurring to us multiple number of times completing the square. So what is completing the square method? So what is the problem with quadratic equation vis-a-vis let's say linear equation? So in a linear equation guys, we had you know very simple way of solving a linear equation in one variable. Let us say we had 5x minus 3 equals 0. So what was the method? Method was just separate the variable side and separate the constant side. So hence we would take this minus 3 on to the right hand side and it would become 5x is equal to 3 and now it becomes very easy to find out x and x is nothing but 3 upon 5. So it was easy for us to do that separation of variables on one side and then you know then solving it by simple cross multiplication. Now problem with quadratic equation is what does a quadratic equation look like? So if you see a quadratic equation would be of the form of ax squared plus bx plus c equals 0 where a is not equal to 0 and a, b and c all belong to the set of real number isn't it? This is quadratic equation. Now even if you separate the variable let's say terms with variables on one side so let me do that. So ax squared plus bx and let me take this c on the other side so it becomes minus c. Now here we have a problem. The problem is there is a x squared term, there is a x term. How do we even find the value of x? What should we do? So at max what you can say you can take x out and it will become ax plus b. So if you take x common and this is equal to minus c but now this is very difficult to find out what is the value of x in this case because if you take one of the terms on the other side for example x would be here minus c upon ax plus b this is what we would have done if it would be a linear equation like here what did we do here? But here if you see there is an x term here as well so we will not be able to find the value of x because the solution itself carries a variable. So what to do? Hence in this one technique could be to linearize the quadratic form or get something in form of two linear terms or something like that where we equate and find the solution what do I mean? So if you had ax squared plus bx plus c somewhat let us say ax squared plus b squared plus c if you somehow reduce it to its factors linear factors. So we discussed it later on earlier also so px plus q and rx plus rx plus s let us say if you somehow do this then also it becomes and then you equate it to 0 and then it will become easier to solve it or let us say if you had ax squared plus bx plus bx equals to minus c what you could do is convert this thing into this form px plus q whole square form this is what is called completing the square form if you do this kind of a thing and here let us say this is you know reduced to k then px px plus q could be written as plus minus root k if you take the square root then you see it becomes a linear equation and now it will be easier to solve this one. So that is what we learn in completing the square method complete this part or the entire quadratic equation or you know convert this term this thing into a square term and the moment square term comes you can take the square root on both sides and basically reduce a quadratic to a linear so if I reduce a quadratic equation to set of linear equations then my job is done so hence this is called this is what is incremental learning so we learned how to solve linear equation previously now we are trying to solve quadratic equation by converting or using the you know principles of solving linear equation right that is what this method elaborates so let us see what how does we how do we do it the quadratic equation would be of the form of ax squared plus bx plus c equals 0 right this is the equation so and we know already that a is not equal to 0 and a b c all belongs to the set of real numbers right now step one what you need to do is you divide the entire equation by a square a divide the equation by a is what coefficient of coefficient of coefficient of x square okay so hence what will you get you will get x square plus b upon ax plus c upon a equals 0 right now what take now here 0 upon so we actually did 0 upon a as well which is equal to 0 okay so remember this now what take the constant constant on RHS take the shift constant term to RHS so what will you get you will get x square plus b by ax is equal to minus c upon a okay third third third step is so add b square upon 4 a square to both sides to both sides and why are we doing this let me show why are we doing this if you see x square plus b by a 2x can be written as 2 times x times b by 2 a right now you know notice this very carefully so this is 2 times x times b times 2 a why because he 2 and 2 is upper integrating cancel out so it will be left with x times b by a which is here so this term is not getting changed but why did I do this arrangement is simply because now if you see this looks like a square terms and here it is 2 a and this item becomes a whole b right so I am saying I am looking at x square as a square right and 2 times a times b where if you see this looks like exactly what I am writing here 2 times a is x and b is b upon 2 a right so if I somehow add here b square what will be b square b square will be simply I am assuming this to be b capital B so it will be b square by 4 a square isn't it so if you add this what will be the advantage this is what is called completing the square so if you notice this is nothing but reduces to a plus b whole square isn't it so that's why I am adding what b square by 4 a square to both sides so if I didn't like this so it will be b square by 4 a square on the right hand side minus c by a was already there correct so now so what will happen so now let us you know fourth step is fourth step will be simplify this and complete the square so complete the square complete the square on LHS so what will it be so if you see what is this square now this thing LHS can be written as x plus b upon 2 a whole squared x plus b by 2 upon a b by b upon 2 a whole squared right LHS is this much you can verify this leads to this only so this is same as this yep and now what on the right hand side let's simplify a little bit so it is 4 a square common denominator and the numerator is b square minus 4 a c so if you see if you take the common denominator 4 a square the numerator will be reduced to b square minus 4 a c correct now what guys so hence now express it as difference of express as difference of 2 square difference of 2 squares and solve what does it mean so first of all we define a term d is called discriminant of quadratic equation discriminant ok where d is equal to b square minus 4 a c so this I am calling it as d you know just a substitution for simplification of calculation as well as other you know purposes so we will see what is the significance of d a little later but for the time being let me call this entire term as capital D so the equation is reduced to x plus b by 2 a whole squared is equal to b by 4 a square ok now the same equation can be written as x plus b by 2 a whole squared minus d by 4 a squared equals 0 and now I can write this as x plus b by 2 a whole squared minus root of d by 2 a whole squared equals 0 right now pause here and understand why did I do this so d can be written as root of d square isn't it so I wrote that there so that I can get a square term here as well so what does this resemble if you see this particular left hand side of this equation resembling a square minus b square form so this is what difference of square form right so difference of square is nothing but a minus b and a plus b isn't it right so I can write this as x plus b by 2 a minus root d by 2 a times x times x plus b by 2 a plus root of plus root of d by 2 a and this is equal to 0 so hence what what can we do now hence what we what can we do now so we can now express this as x plus b by 2 a minus root d by 2 a equals to 0 or x plus b by 2 a plus root d by 2 a equals to 0 right we have seen that by factorization method solving the equation by factorization method so from here what do we get we get x is equal to minus b by 2 a plus root d by 2 a or from here you'll get x is equal to minus b by 2 a minus root d by 2 a isn't it right so hence you can write this as minus b plus you know root d is nothing but b square minus 4 a c upon twice of a or x is equal to minus b minus root of b square minus 4 ac upon twice of a correct so you will see this is also the method of you know finding the roots of a quadratic equation by you know quadratic formula so eventually this is what is called quadratic formula as well quadratic formula as well so what are the values so let me summarize summary so summary is very simple so if you have a quadratic equation ax square plus bx plus c equals 0 then where a is not equal to 0 and a b and c are real real real numbers then x solution x is equal to minus b minus under root b square minus 4 ac by twice a and x is equal to minus b plus under root b square minus 4 ac by 2 a please remember this yeah this is the you know if any any time you get stuck while solving a quadratic equation this formula is going to rescue you so this is valid under all circumstances right so a not equals to 0 if you see here in the denominator is to a right so clearly a cannot be 0 otherwise there are no roots defined isn't it so hence a cannot be 0 from this perspective as well let us take an example example could be let us say I have an equation 9x square minus 15x minus 15x plus 6 equals 0 so what are the values of a and b and c a is 9 b is 15 and sorry b is minus 15 do not so this is a mistake many people do including me as well so minus 15 and c is equal to 6 so never never miss the sign now what so let us use the formula above which we derived so x will be equal to what minus b so minus b is minus minus 15 then minus under root b square so minus 15 whole square minus 4 times a a is 9 and c is 6 c is 6 divided by twice of a that is 2 into 9 right so what will this value be this is nothing but 15 minus under root 225 minus this is nothing but 4936 and 36 into 6 is 216 so if you see this is 216 right divided by 18 18 and it is 15 minus under root 225 minus 216 is 9 divided by 18 so hence it is 15 minus 3 upon 18 which is 12 upon 18 hence it is 2 upon 3 correct one value is 2 upon 3 and another value will be simply you have to instead of this minus there will be a plus so hence another value is simply 15 plus root 9 upon 18 so hence it is nothing but 15 plus 3 18 upon 18 that is 1 so if you check these are the two two solutions two solution to the given quadratic equation but we had to do it by completing the square we just check the formula the time formula definitely works but how to come you know solve it using completing the square method let us do it again 9 x square minus 15 x plus 6 equals 0 what was the first step first step was if you see divide the entire thing by a so hence you will get x square minus 15 upon 9 a is 9 here x plus 6 upon 9 equals 0 then what do you do you take the constant term onto the right hand side so 15 by 9 x is equal to minus 6 upon 9 then you do what you add b square minus 4 a b square sorry you add a b what is that b square by 4 square to both sides so hence you add x square minus 15 by 9 or you can if you don't want to mug up the steps so what you can do is you can do step by step which can indicate you all that so x square you can write minus 2 times x times 15 upon 2 times 9 that is what we did and this is minus and I can you know simplify 6 by 9 and write 2 upon 3 okay now this will get give you a hint of completing the square how to complete the square x square minus 2 times x times 15 upon 18 and then you add 15 square by 18 square is it it so that's this is what the b term was so I added that so I have to add both sides 15 square by 18 square and now 2 by 3 was as it is right now this will complete my square so you can write this as x minus 15 upon 18 whole squared right a minus b whole square is a square minus 2 a b plus b square which was above so I completed the square and this is nothing but now let us simplify 15 upon 18 is 5 upon 6 isn't it 15 upon 18 is 5 upon 6 so it is 5 square by 6 square minus 2 by 3 okay so what will this be now this is nothing but x minus 5 upon 6 whole square is equal to now this is 25 by 36 and I can you know equate the denominators so make the denominator same so it will be 24 upon 36 if you see 2 upon 3 is 24 upon 36 so hence this becomes x minus 5 by 6 whole square is equal to 1 upon 36 isn't it so hence if you really you know take the square root both the sides or you can do the difference method so you can say x minus 5 by 6 whole square minus 1 by 6 whole square is equal to 0 I took 1 by 36 on the left hand side and represented is as 1 by 6 whole square and now difference of 2 square will be x minus 5 by 6 minus 1 by 6 a minus v term and a plus b term will be x minus 5 by 6 plus 1 by 6 right and this is equal to 0 because why if you consider this as this whole as a so this is a square and this is b square so hence next step will be a minus b a plus b equals 0 right and that is what was there in this step so simplifying it further or now taking you know 1 by 1 all the factors can be equated to 0 so but let's simplify first so this is if you see the first term is nothing but minus 6 upon 6 and this one is x minus 4 upon 6 is equal to 0 so hence x equals to 6 upon 6 that is 1 or x equals to 4 upon 6 which is equal to 2 upon 3 so if you check we got the same result with complete by completing the square method as well and when we deployed the formula then also we got the same result 2 by 3 here and 1 here so we will take up more of such problems solved examples in the subsequent sessions