# How do you factor 2x^2 -7x -15?

May 27, 2016

$\left(x - 5\right) \left(2 x + 3\right)$

Thought processes explained!

#### Explanation:

Whole number factor of 2 can only be {2 , 1}

Whole number factors of 15 are {1 , 15}; {3 , 5}
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Lets experiment!

We know that we have to start with: (1x+?)(2x+?)
Which when written as per convention is (x+?)(2x+?)

We know that $2 \times 15 = 30$ which is a long way from the 7 in $7 x$ so we will not use that to start with

'....................................................
Try this:

(x+3)(2x+5) = 2x^2+5x+6x+15 " "color(red)("Fail!")
For a start we have +15 and we need -15. Also the 7 from $7 x$ is -7. So the larger of the two gas to be negative.

'.................................................
Try this:

(x+3)(2x-5) = 2x^2-5x+3x-15 " "color(red)("Fail. This gives -2x")

'.........................................
Try swapping the 3 and 5 around

(x-5)(2x+3) = 2x^2+3x-10x-15 " "color(green)("Works!")

$2 {x}^{2} + 3 x - 10 x - 15 \text{ "=" } 2 {x}^{2} - 7 x - 15$