Question

Find the value of c that makes the trinomial a perfect square trinomial x^2+8x+c

Answer 1

To determine the value of \( c \) that makes the trinomial \( x^2 + 8x + c \) a perfect square trinomial, we can use the formula for a perfect square trinomial:

\[
(x + a)^2 = x^2 + 2ax + a^2
\]

In our case, we need to match the form \( x^2 + 2ax + a^2 \) to \( x^2 + 8x + c \).

Here, we identify that \( 2a = 8 \). To find \( a \), we solve for \( a \):

\[
a = \frac{8}{2} = 4
\]

Now that we have \( a = 4 \), we can find \( c \) by substituting \( a \) into the term \( a^2 \):

\[
c = a^2 = 4^2 = 16
\]

Thus, the value of \( c \) that makes the trinomial \( x^2 + 8x + c \) a perfect square trinomial is

\[
\boxed{16}
\]

Verifying, we can rewrite the trinomial as:

\[
x^2 + 8x + 16 = (x + 4)^2
\]

This confirms that \( x^2 + 8x + 16 \) is indeed a perfect square trinomial.