To determine the value of [tex]\( c \)[/tex] that makes [tex]\( x^2 + 6x + c \)[/tex] a perfect square trinomial, we need to express it in the form [tex]\((x + a)^2\)[/tex], where [tex]\(a\)[/tex] is a constant.
1. Expand [tex]\((x + a)^2\)[/tex]:
[tex]\[
(x + a)(x + a) = x^2 + 2ax + a^2
\][/tex]
2. Compare this with the given expression [tex]\( x^2 + 6x + c \)[/tex]. We see that:
[tex]\[
x^2 + 6x + c = x^2 + 2ax + a^2
\][/tex]
3. By matching the coefficients of [tex]\(x\)[/tex], we have:
[tex]\[
2a = 6
\][/tex]
4. Solve for [tex]\(a\)[/tex]:
[tex]\[
a = \frac{6}{2} = 3
\][/tex]
5. Now, plug the value of [tex]\(a\)[/tex] back into [tex]\( a^2 \)[/tex] to find [tex]\( c \)[/tex]:
[tex]\[
c = a^2 = 3^2 = 9
\][/tex]
Thus, the value of [tex]\( c \)[/tex] that makes [tex]\( x^2 + 6x + c \)[/tex] a perfect square trinomial is [tex]\( \boxed{9} \)[/tex].
