Question

How do you factor `2x^2-x-15`?

Answer 1

`(2x + 5)(x - 3)`

Lol I hope you aren't trying to find your homework answers online to copy!

Explanation:

Firstly, consider the first term. Since it's `2x^2`, there are only two linear factors that it can be 'broken down' into: `x` and `2x`. Thus we can re-write the expression `2x^2 - x - 15` as `(2x + a)(x + b)` where `a` and `b` are real numbers to be determined

Thus
`2x^2 - x - 15 = (2x + a)(x + b)`
`2x^2 - x - 15 = 2x^2 + (a + 2b)x + ab`

Consider the last term. In order to get a negative number, we need one positive and one negative number. Thus, if `a` is positive, `b` is negative and vice versa.

Now lets think about the factors that make up `15`.
`15 = 1 * 15 = 3 * 5`

Now guess-and-check (not much to do) which combination (`1` and `15` or `3` and `5`) will give `-1` (coefficient of center term), that is, which `a` and `b` will fit the expression `a + 2b = -1`. Remember that if `a` is positive, `b` is negative and so on. With some luck you should get:

`5 + 2(-3) = -1`, `a = 5`, `b = -3`

Thus,
`2x^2 - x - 15 = (2x + a)(x + b) = (2x + 5)(x - 3)`