To determine the value of \( c \) that makes the expression \( x^2 - 12x + c \) a perfect square trinomial, we need to consider what it means for a quadratic expression to be a perfect square trinomial. A perfect square trinomial is of the form \( (x + a)^2 \), which expands to \( x^2 + 2ax + a^2 \).
Given the expression \( x^2 - 12x + c \), we can compare it to the general form \( x^2 + 2ax + a^2 \). Notice that the term \( -12x \) corresponds to \( 2ax \) in the perfect square form.
To find the value of \( a \):
[tex]\[ 2a = -12 \][/tex]
Dividing both sides by 2:
[tex]\[ a = -6 \][/tex]
Now, we find \( c \) by squaring \( a \):
[tex]\[ c = a^2 \][/tex]
[tex]\[ c = (-6)^2 \][/tex]
[tex]\[ c = 36 \][/tex]
Therefore, the value of \( c \) that makes the expression \( x^2 - 12x + c \) a perfect square trinomial is \( 36 \).
So, the correct value of \( c \) is:
[tex]\[ 36 \][/tex]
