Question

How do you factor `8x^2 + 28x + 12` completely?

Answer 1

`(4x+2)(2x+6)`

Explanation:

With quadratic equations, the key is to think about the factors of each section, then work out which ones go together.

`8x^2` can be produced by:
`8x xx1x` or `-8x xx-1x`
`2x xx 4x` or `-2x xx-4x`

`12` can be made by:
`1xx12` or `-1xx-12`
`2xx6` or `-2xx-6`
`3xx4` or `-4xx-3`

When we look at these factors, we have to think how we can multiply the `8x^2` factors by the `12` factors to make `28x` when added.

In this case, we take `2` and `4` then `2` and `6`.
`4x xx6 = 24x` and `2x xx2=4x`
`24x+4x=28x`

So we write this as:
`(4x+2)(2x+6)`

To check this, just expand it out again!
`4x xx 2x = 8x^2`
`4x xx 6 = 24x`
`2 xx 2x = 4x`
`2 xx 6 = 12`
`8x^2 + 24x + 4x + 12 = 8x^2 + 28x+ 12`
So we know it's correct!

Hope this helps; let me know if I can do anything else for you:)

Answer 2

`y = 4(2x + 1)(x + 3)`

Explanation:

y = 4(2x^2 + 7x + 3) = 4Y = 4(2)(x + p)(x + q)
Factor Y by the new AC Method (Socratic Search)
Converted Y' = x^2 + 7x + 6.
Factor pairs of ax = 6 --> (1, 6). This sum is 7 = b.
Therefor `p = 1/2` and `q = 6/2 = 3`.
Factored form:
`y = 4(2)(x + 1/2)(x + 3) = 4(2x + 1)(x + 3)`