Question

Is it possible to factor `y=3x^2+11x-4`? If so, what are the factors?

Answer 1

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

Explanation:

Use the AC method.

`y=3x^2+11x-4` is a quadratic equation in the form `ax+bx+c`, where `a=3, b=11, and c=-4`.

Multiply `a` times `c`.

`3xx-4=-12`

Find two numbers that when added equal `11` and when multiplied equal `-12`. The numbers `12` and `-1` meet the criteria.

Rewrite the equation substituting `12x` and `-x` for `11x`.

`3x^2+12x-x-4`

Group the terms into two groups of two terms.

`(3x^2+12x)-(x-4)`

Factor out `3x` from the first group and `-1` from the second group.

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

Factor out `(x+4)`.

`(x+4)(3x-1)`

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

Answer 2

`color(brown)((3x-1)(x+4))`

Explanation:

3 is prime so we can only have 1 and 3 as factors giving:

`(x+?)(3x+?)`

There are two sets of factors of 4 and they are: { 2,2} and {1,4}

Lets try the {1,4} combination and see what we get!

By the way, the constant of 4 in your question is negative so the `color(white)(....)`{1 ,4} have to be opposite in their sign.

We see that `3xx4=12` which is close to the `11` in `11x` so lets try that configuration:

`(x-4)(3x+1) = 3x^2-12x+x-4` Not quite correct as we have `-11x`.

Ok! Lets try something else. The 11 is the correct magnitude but the wrong sign. Lets try reversing the signs:

`(x+4)(3x-1) =3x^2+12x-x-4` Now we have it!

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So `color(white)(..)3x^2+11x-4 = color(brown)((x+4)(3x-1))`

Or if you like you can change the order of the brackets to.

`color(brown)((3x-1)(x+4))`

`color(blue)("They call this Commutative")`

in that they can commute/travel without changing the intrinsic value