Let's solve the problem step-by-step.
1. Original Equation:
[tex]\[
x^2 - \frac{3}{4}x = 5
\][/tex]
2. Identify the coefficient of [tex]\(x\)[/tex]:
The coefficient of [tex]\(x\)[/tex] is [tex]\(-\frac{3}{4}\)[/tex].
3. Determine the value to add:
To transform the left side of the equation into a perfect-square trinomial, we need to add [tex]\(\left(\frac{b}{2}\right)^2\)[/tex] to both sides, where [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex].
4. Calculate [tex]\(\left(\frac{b}{2}\right)^2\)[/tex]:
Here, [tex]\(b = -\frac{3}{4}\)[/tex]. Thus,
[tex]\[
\left(\frac{b}{2}\right)^2 = \left(\frac{-\frac{3}{4}}{2}\right)^2
\][/tex]
5. Simplify the calculation:
[tex]\[
\left(\frac{-\frac{3}{4}}{2}\right)^2 = \left(-\frac{3}{4} \cdot \frac{1}{2}\right)^2 = \left(-\frac{3}{8}\right)^2 = \frac{9}{64}
\][/tex]
6. Conclusion:
The value that must be added to both sides of the equation to make the left side a perfect-square trinomial is [tex]\(\frac{9}{64}\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{\frac{9}{64}}
\][/tex]
