Complete the square and factor the resulting perfect-square trinomial.

Question

Quizplus · Accepted Answer

We can complete the square as follows:
$8t^2+11t-15 = 8\left(t+\frac{11}{16}\right)^2 - \frac{121}{32} - 15$
$= 8\left(t+\frac{11}{16}\right)^2 - \frac{481}{32}$
So we have a perfect-square trinomial with $a=8$, $b=\frac{11}{16}$, and $c=-\frac{481}{32}$. To factor this, we can use the form $(a+b)^2 = a^2 + 2ab + b^2$ with $a=2\sqrt{2}t$ and $b=\frac{11\sqrt{2}}{8}$:
$8\left(t+\frac{11}{16}\right)^2 - \frac{481}{32} = 8\left(\left(\sqrt{2}t\right)^2 + 2\left(\sqrt{2}t\right)\left(\frac{11\sqrt{2}}{8}\right) + \left(\frac{11\sqrt{2}}{8}\right)^2\right) - \frac{481}{32}$
$= 8\left(2t+\frac{11\sqrt{2}}{4}\right)^2 - \frac{481}{32}$
So the correct choice is (C).

Complete the Square and Factor the Resulting Perfect-Square Trinomial