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].
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].
