Question

How do you factor the trinomial `x^2 - 14x + 24`?

Answer 1

The trinomial `x^2-14x+24` can be factored as `(x-2)(x-12)`.

Explanation:

Approach 1
Since both `-14` and `24` are relatively small and easy numbers. We can just look for two numbers `a,b` that satisfy the following two equations:

`ab=24`
`a+b=-(-14)=14`.

After some experimentation, we arrive at our answer.

Approach 2
For quadratic equations in general, it often makes sense to simply go straight to the quadratic formula to solve for `x` directly.

It gives you `x=7+-5`. It is trivial to figure out the factorization from here.

Approach 3

We use

`alpha + beta = -b/a`
`alpha * beta = c/a`

In general we can express an equation as

`(x-alpha)(x-beta)`

so
`14 = alpha + beta`
`24 = alphabeta`

Now you can solve these 2 equation for `alpha` and `beta`

But on close inspection we find

`alpha = 12, beta = 2`

So the above equation is

`(x-12 )(x-2)`