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

Dec 14, 2015

Yes; $y = {\left(x + 6\right)}^{2}$

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:

$1 , 36 \to 37$
$2 , 18 \to 20$
$3 , 12 \to 15$
$4 , 9 \to 13$
$6 , 6 \to 12$

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

$\left(x + 6\right) \left(x + 6\right) = {\left(x + 6\right)}^{2}$

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:

${\left(a + b\right)}^{2} = {a}^{2} + 2 a b + {b}^{2}$

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

${\left(x + 6\right)}^{2} = {x}^{2} + 2 \left(6\right) \left(x\right) + {6}^{2} = {x}^{2} + 12 x + 36$

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

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

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