Question

How do you solve `3x^2 + 9x = 12`?

Answer 1

`" "(3x-3)(x+4)`

With practise these become easier to solve

Explanation:

Given: `" "3x^2+9x=12`

Write as:

`" "3x^2+9x-12=0`

We are looking for `(?x+?)(?x+?)`

The only whole number factors of 12 are
`color(blue)(" "1 ,12)`
`color(blue)(" "2,6)`
`color(blue)(" "3,4)`

This means we can only have one of:
`" "(?x+1)(?x+12)`
`" "(?x+2)(?x+6)`
`" "(?x+3)(?x+4)`

'................................................................

The only whole number factors of 3 are `color(magenta)(1,3)`

So the `?x` can only be these numbers giving:

`" "(color(magenta)(1)x+?)(color(magenta)(3)x+?) -> (x+?)(3x+?)`
'..............................................................

It is now a matter of trying them out and seeing what works.

I spot that from the factors of `color(magenta)(3)` if we have:

`color(magenta)((3color(blue)(xx4))-(1color(blue)(xx3)))`

we have:`" "12-3=9` which will give us `+9x`

So the factorisation must be

`" "(color(magenta)(3x)color(blue)(-3))(color(magenta)(x)color(blue)(+4))`

'~~~~~~~~~~~~~~~~~~~~~
Check

`" "(3x-3)(x+4)`

`" "=" "3x^2+12x-3x-12`
`" "=" "3x^2+9x-12`