Question

For what real values of $c$ is $x^2 + 16x + c$ the square of a binomial? if you find more than one, then list your values separated by commas.

Answer 1

The Square of a Binomial is always a trinomial. In general, you have the following relationship for a square binomial:

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

The trinomial of our problem is:

$x^2+16x+c$

So we need to find the real value of $c$ such that this is a square of a binomial. Matching the general form with the trinomial of our problem we have the following equations:

$a=x \\ 2ab=16x = 2(x)(8) \\ where \ b=8 \\ \\b^2=c \therefore c=8^2 \therefore \boxed{c=64}$

Therefore the square of our binomial is:

$(x+8)^2=x^2+16x+64$