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

1 Answer
Dec 14, 2015

Answer:

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.