If $a$ is a constant such that $4x^2 - 12x + a$ is the square of a binomial, then what is $a$?
I want to find a constant $a$ that makes $4x^2 - 12x + a$ a perfect square trinomial. That means I want to write $4x^2 - 12x + a$ as $(bx + c)^2$, where $b$ and $c$ are some constants. If I expand $(bx + c)^2$, I get $b^2 x^2 + 2bcx + c^2$. Comparing this with $4x^2 - 12x + a$, I see that $b^2 = 4$, so $b = \pm 2$. I also see that $2bc = -12$, so $bc = -6$. Since $b = \pm 2$, I can divide both sides by $b$ to get $c = \mp 3$. Finally, I see that $c^2 = a$, so $a = (\mp 3)^2 = 9$. 