To determine which value must be added to the expression [tex]\(x^2 - 3x\)[/tex] to make it a perfect-square trinomial, follow these steps:
1. Identify the coefficient of [tex]\(x\)[/tex]:
In the expression [tex]\(x^2 - 3x\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-3\)[/tex].
2. Calculate half of the coefficient of [tex]\(x\)[/tex]:
Take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-3\)[/tex], and divide it by 2:
[tex]\[
\frac{-3}{2} = -\frac{3}{2}
\][/tex]
3. Square the result:
Square [tex]\(-\frac{3}{2}\)[/tex] to find the value that needs to be added:
[tex]\[
\left( -\frac{3}{2} \right)^2 = \frac{9}{4}
\][/tex]
Therefore, the value that must be added to the expression [tex]\(x^2 - 3x\)[/tex] to make it a perfect-square trinomial is [tex]\(\frac{9}{4}\)[/tex].
So, the correct answer is:
[tex]\[
\boxed{\frac{9}{4}}
\][/tex]
