To find the missing value \( c \) that makes the expression \( x^2 - 26x + c \) a perfect square trinomial, follow these steps:
1. Identify the general form of a perfect square trinomial: A perfect square trinomial takes the form \((x - a)^2\). Expanding this form, we get:
[tex]\[
(x - a)^2 = x^2 - 2ax + a^2
\][/tex]
2. Compare terms with the given expression: In our given expression \( x^2 - 26x + c \), we can compare it with the expanded form of the perfect square trinomial \( x^2 - 2ax + a^2 \).
By comparing the coefficients:
[tex]\[
x^2 - 26x + c \quad \text{and} \quad x^2 - 2ax + a^2
\][/tex]
We notice that the coefficient of the linear term (\(-26\)) should match the coefficient of \(-2ax\).
3. Determine \( a \): Set the coefficients of the linear terms equal to each other:
[tex]\[
-2a = -26
\][/tex]
Solving for \( a \):
[tex]\[
2a = 26 \quad \Rightarrow \quad a = 13
\][/tex]
4. Find the value of \( c \): The value of \( c \) in the perfect square trinomial is given by \( a^2 \).
So,
[tex]\[
c = a^2 = 13^2 = 169
\][/tex]
Therefore, the value of [tex]\( c \)[/tex] that makes [tex]\( x^2 - 26x + c \)[/tex] a perfect square trinomial is [tex]\(\boxed{169}\)[/tex].
