Answer :

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].