To make the expression [tex]\(x^2 + 16x\)[/tex] a perfect-square trinomial, we need to find a value that, when added to this expression, allows it to be factored as a square of a binomial.
A perfect-square trinomial has the form:
[tex]\[
(a + b)^2 = a^2 + 2ab + b^2
\][/tex]
For the expression [tex]\(x^2 + 16x\)[/tex], the [tex]\(a^2\)[/tex] part corresponds to [tex]\(x^2\)[/tex] and the [tex]\(2ab\)[/tex] part corresponds to [tex]\(16x\)[/tex].
First, we identify the coefficients:
- The coefficient of [tex]\(x\)[/tex] is 16.
Now, recall the general form where [tex]\(2ab\)[/tex] (in our case [tex]\(2 \cdot x \cdot b\)[/tex]) matches the term [tex]\(16x\)[/tex]. To define [tex]\(b\)[/tex], we set up the equation:
[tex]\[
2 \cdot x \cdot b = 16x
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
2b = 16
\][/tex]
[tex]\[
b = \frac{16}{2} = 8
\][/tex]
Next, to convert [tex]\(x^2 + 16x\)[/tex] into a perfect-square trinomial, we add [tex]\(b^2\)[/tex] to the expression:
[tex]\[
b^2 = 8^2 = 64
\][/tex]
Therefore, the value that must be added to [tex]\(x^2 + 16x\)[/tex] to make it a perfect-square trinomial is:
[tex]\[
\boxed{64}
\][/tex]
