Sure! To create a perfect square trinomial from the quadratic expression \( x^2 + 16x + c \), we need to determine a value of \( c \) that completes the square. Here’s the step-by-step process:
1. Identify the coefficients: For the quadratic expression \( x^2 + 16x + c \), we have:
- The coefficient of \( x^2 \) (denoted as \( a \)) is 1.
- The coefficient of \( x \) (denoted as \( b \)) is 16.
2. Formula for the perfect square trinomial: A perfect square trinomial takes the form \( (x + d)^2 \), which expands to \( x^2 + 2dx + d^2 \).
3. Compare coefficients: We need the expression \( x^2 + 16x + c \) to match the form \( x^2 + 2dx + d^2 \). The coefficient of \( x \) in our expression is 16, which should match \( 2d \) in the perfect square trinomial:
[tex]\[
2d = 16
\][/tex]
4. Solve for \( d \):
[tex]\[
d = \frac{16}{2} = 8
\][/tex]
5. Calculate \( c \): To complete the square, we need \( c \) to be \( d^2 \):
[tex]\[
c = d^2 = 8^2 = 64
\][/tex]
6. Form the perfect square trinomial: Substitute \( c = 64 \) back into the expression:
[tex]\[
x^2 + 16x + 64
\][/tex]
Thus, the value of \( c \) needed to create a perfect square trinomial is 64, and the resulting perfect square trinomial is:
[tex]\[
x^2 + 16x + 64
\][/tex]
This trinomial can be written as the square of a binomial:
[tex]\[
(x + 8)^2
\][/tex]
So, the perfect square trinomial formed is [tex]\( x^2 + 16x + 64 \)[/tex].
