Question

Is it possible to factor `y=x^2 + 12x + 36`? If so, what are the factors?

Answer 1

Yes; `y = (x+6)^2`

Explanation:

There are multiple ways to factor trinomials, so let's begin by asking:

What two numbers multiply to give 36 and add to give 12?

There are only a handful of numbers that multiply to give 36, so the guess and check methods isn't a bad idea. Let's write the possible factors of 36 and find their sum:

`1,36 -> 37`
`2,18 -> 20`
`3,12 -> 15`
`4,9 -> 13`
`6,6 -> 12`

Looks like we found a pair! So, the factorization becomes:

`(x+6)(x+6) = (x+6)^2`

Another way to do this is knowing a special case of trinomials called perfect square trinomials. All trinomials that are a perfect square of a binomial are of the form:

`(a+b)^2 = a^2 + 2ab + b^2`

In this case, `a=x` and `b=6`, so:

`(x+6)^2 = x^2 + 2(6)(x) + 6^2 = x^2 + 12x + 36`

We can also double check our answer by comparing the graphs:

`y=x^2+12x+36`
graph{x^2+12x+36 [-10, 10, -5, 5]}
`y=(x+6)^2`
graph{(x+6)^2 [-10, 10, -5, 5]}

They are identical, confirming we have found the right answer.