Hello masterinex Originally Posted by

**masterinex** I guess the idea is like this :

9x^2 -12x +4

Sum = -12

product = 36

a and b are just two numbers which added together give -12 and multiplied get 36 . Then the linear term -12x can be rewritten in terms of ax + bx

a= -6 b=-6

so 9x^2 -12x +4 becomes

= 9x^2 -6x-6x+4

(3x-2)(3x-2)

for this example the method worked fine . But what about if I have something like this :

2x^2 + 11x + 5 for which the factored form should be (2x+1)(x+5)

Then by the method

product = 10 and sum = 11

the only two numbers I can think of are 2 and 5 , but the sum wouldnt be 11. Any suggestions ?

You'll kick yourself - what about $\displaystyle 10$ and $\displaystyle 1$?

Then you can say (using the same method as before):

$\displaystyle 2x^2 + 11x + 5 = 2x^2+10x+x+5$$\displaystyle =2x(x+5)+1(x+5)$

$\displaystyle =(2x+1)(x+5)$

Any quadratic that *can *be factorised will work in the same way, even with numbers with a large number of factors. For example:

$\displaystyle 45x^2+126x-80$

Product $\displaystyle = 45 \times (-80) = -3600$. So we want two factors of $\displaystyle 3600$ whose difference is $\displaystyle 126$. A certain amount of trial and error is necessary, but $\displaystyle 150$ and $\displaystyle 24$ are the factors required. So:$\displaystyle 45x^2+126x-80$

$\displaystyle =45x^2 +150x-24x-80$

$\displaystyle =15x(3x+10)-8(3x+10)$

$\displaystyle =(15x-8)(3x+10)$

Grandad