Question

Solve using the square root property.

$64a^2+48a+9=81$

Short answer

The solution is$a=\frac{3}{4}$and$a=-\frac{3}{2}$.

Step-by-step solution

Step 1. Given information.

The given equation is$64a^2+48a+9=81$.

Step 2. Apply the square root property.

The property states that if$x=k^2$, then$x=\pm \sqrt{k}$.

Step 3. Simplify the equation.

$\begin{array}{rcl}64a^2+48a+9& =& 81\\ {\left(8a\right)}^2+2·8a·3+{\left(3\right)}^2& =& 81\\ {\left(8a+3\right)}^2& =& 81\\ 8a+3& =& \pm \sqrt{81}\\ 8a+3& =& \pm 9\end{array}$

Step 4. Solve the equation for a.

$\begin{array}{rcl}8a+3& =& 9\\ 8a& =& 9-3\\ 8a& =& 6\\ a& =& \frac{6}{8}\\ a& =& \frac{3}{4}\end{array}$androle="math" localid="1645251828054" $\begin{array}{rcl}8a+3& =& -9\\ 8a& =& -9-3\\ 8a& =& -12\\ a& =& \frac{-12}{8}\\ a& =& \frac{-3}{2}\end{array}$

Step 5. Simplified answer.

Hence, the required solution is $a=\frac{3}{4}$and$a=-\frac{3}{2}$.