 in this question we have been asked to solve the given quadratic equation by factorization method so another one of such type where the coefficients are non-numerals okay so hence what is the methodology we have to use factorization techniques using algebraic identities or otherwise to split the middle term so if you see here this equation is of the form of ax square plus bx plus c equals 0 where you know this is c this entire term within brackets is c minus 9a plus b is b and a is 9 so hence our a c is a times c is 9 into 2a square plus 5ab plus 2b square now you have to split the b term this b term such that b is equal to b1 plus b2 such that b1b2 must be equal to ac correct that means I have to first factorize ac so let's first factorize ac so if you see ac what is a times c so it is 9 into twice a square plus 5ab plus twice b square this itself will require a splitting the middle term to factorize and we have learned that already so how what should we do so it is nothing but 9 2a square and this 5ab can be written as 4ab plus a b plus 2b square the remaining 2b square why did I do that because if you see 4 plus 1 is 5 certainly and 4 into 1 is 4 right so this 2 and 2 multiplied together was giving you 4 so hence I split the middle term into 4 and 1 so that the product of these two is the product of these two correct and the sum 4ab plus a b is 5ab correct so hence this will become 9 into now take 2a common so if you take 2a common you'll get 2a plus 2b 2a sorry you will get a plus 2b you will get a plus 2b and then here if you take a common again you'll get a plus sorry here sorry here you'll take b common b common so if you take b common you will get a plus 2b again correct so hence factorization will be 9 times 2a plus b times a plus 2b these are the factors okay now if you see if you if you if you see what can what can we said about the middle term now so these are the factors these are the factors and if you yes so what you can what you can do is let us say we have 9a plus b there as the b right so basically this particular thing can be broken down as 3 times 2a plus b and this can be broke and the other term is 3 times a plus 2b is it if you take these two as factors together let us say two one factor is this whole another factor is this whole so let us say we can we are I'm saying I can treat it as b1 and b2 now let's check what is b1 plus b2 so we even plus b2 if you see it's nothing but 3 times 2a plus b plus 3 times a plus 2b which is nothing but 6a plus 3b plus 3a plus 6b which is nothing but 9a plus 9b which is nothing but 9a plus b so you see b1 plus b2 is nothing is giving out you sum them up you are getting b here so that's what we wanted so let's solve them so what will it be then it will be 9x square and I can write minus 9a plus 9b can be written as 3 times 2a plus bx minus 3 times a plus 2bx that's what is the middle term so basically I broke the middle term into two such that their product is 9 times 9 times the constant term which I'm going to write now so the constant term is this what is the constant term in 2a plus b times a plus 2b and this is equal to 0 now the entire equation has been has been you know reduced to this beautiful looking form right so again reemphasizing I split the middle term into two terms here and where did I get this idea from so basically I performed a times c and then factorize it and broke that factorization in such a way that the sum is the middle term that's what I did right so now you can take comments and the equation is solved so if you take common you'll get 3x common from both these first two factors and it will be x minus 2a plus b and and in the second the last two factors you can take you can take sorry there will be three here yeah three here left oh I'm sorry no not three here three will be here 3x minus something right 3x minus 2a plus b and now if you take what will be common here it will see a plus 2b is common a plus 2b is common and you will get what 3x minus minus 2a plus b again and this is equal to 0 so further factorization will lead to 3x minus 2a plus b multiplied by 3x minus a plus 2b and this is equal to 0 so again equating each of the factor to 0 this will be 0 and this will be 0 while linear factors must be equal to 0 for the product to be 0 so hence you will get 3x minus 2a plus b equals 0 or 3x minus a plus 2b is equal to 0 okay so hence from here you'll get x is equal to 2a plus b or x is equal to a plus sorry this will be 2a plus b by 3 and here a plus 2b upon 3 these are the two solution to this question dear friends so learning is even if the coefficients are non-numeral you can perform the splitting the middle term methodology use that methodology and split the middle term factorize a and c so that you know you get two terms whose sum is b and that's what we performed here and eventually got the solution