To solve the equation [tex]\( x^2 - \frac{3}{4} x = 5 \)[/tex] by completing the square, we need to transform the left side into a perfect square trinomial of the form [tex]\( (x - b)^2 \)[/tex].
Here are the steps to do this:
1. Identify the coefficient of the linear term: In the given equation [tex]\( x^2 - \frac{3}{4} x = 5 \)[/tex], the coefficient of [tex]\( x \)[/tex] is [tex]\( -\frac{3}{4} \)[/tex].
2. Calculate half of the coefficient: Half of [tex]\( -\frac{3}{4} \)[/tex] is [tex]\( \frac{-3/4}{2} \)[/tex].
[tex]\[
\frac{-3/4}{2} = -\frac{3}{8}
\][/tex]
3. Square the result of step 2: Now, we square [tex]\( -\frac{3}{8} \)[/tex].
[tex]\[
\left( -\frac{3}{8} \right)^2 = \left( \frac{3}{8} \right)^2 = \frac{9}{64}
\][/tex]
By squaring half of the coefficient of [tex]\( x \)[/tex], we determined that the value to add to both sides of the equation to make the left side a perfect-square trinomial is [tex]\( \frac{9}{64} \)[/tex].
Thus, Brian must add [tex]\(\frac{9}{64}\)[/tex] to both sides of the equation. The correct choice is:
[tex]\[
\boxed{\frac{9}{64}}
\][/tex]
