Question

How do you factor `18x^2-9x-14`?

Answer 1

`(3x+2)(6x-7)`

Explanation:

The factors of 18 are (1 and 18), (-1 and -18), (2 and 9), (-2 and -9), (3 and 6), and (-3 and -6).

So, it could either be:
`(x+a)(18x+b)`,
`(-x+a)(-18x+b)`,
`(2x+a)(9x+b)`,
`(-2x+a)(-9x+b)`,
`(3x+a)(6x+b)`,
`(-3x+a)(-6x+b)`.

The factors of -14 are (1 and -14), (-1 and 14), (-2 and 7), or (2 and -7)

Now we need to combine a set of factors of -14 and 18 that would add to give -9.

These are:
-14 - (-2 and 7)
18 - (-3 and -6)

`(-2*-6)+(-3*7) = 12-21 = -9`

So, `18^2-9x-14-=(-3x-2)(-6x+7)`.

Dividing a negative out of each bracket gives the factors as:

`(3x+2)(6x-7)`

Answer 2

`(6x-7)(3x+2)`

Explanation:

Find factors of `18 and 14` whose products differ by 9.

This involves a bit of trial and error to find them.

`" "18" "14`
`" "darr" "darr`

`" "6" "7 " "rarr 3xx7 = 21`
`" "3" "2" "rarr 6xx2 = ul12`
`color(white)(wwwwww...wwwwwwwww)9`

These are the correct factors, now look at the signs.
The difference needs to be `-9`

`" "6" "-7 " "rarr 3xx-7 = -21`
`" "3" "+2" "rarr 6xx+2 = +ul12`
`color(white)(wwwwww...wwwwwwwwwwwww)-9`

This gives the factors as:

`(6x-7)(3x+2)`

An alternative method is to multiply `18 and 14` and find factors which differ by `9`.

`18 xx14 = 252`

Find factors which are close to the square root of `252`
`sqrt252 = 15.87`

A difference of `9` means that there will be about `4` or `5` either side of `15.87`
A bit of trial and error results in `12 xx21 = 252`
This gives us a clue as to the factors we need to work towards.