To find the value of \( c \) that makes the expression \( x^2 - 24x + c \) a perfect square trinomial, we need to complete the square.
A trinomial of the form \( x^2 + bx + c \) is a perfect square trinomial if it can be written as \( (x + d)^2 \). When expanded, this equals:
[tex]\[
(x + d)^2 = x^2 + 2dx + d^2
\][/tex]
By comparing \( x^2 + 2dx + d^2 \) with \( x^2 - 24x + c \):
- We see that \( 2d \) must equal \(-24\).
Thus:
[tex]\[
2d = -24
\][/tex]
Solving for \( d \):
[tex]\[
d = -12
\][/tex]
Now we need to find \( c \). The value of \( c \) is \( d^2 \), which is:
[tex]\[
c = (-12)^2 = 144
\][/tex]
Thus, the value of \( c \) that makes \( x^2 - 24x + c \) a perfect square trinomial is \( 144 \).
So, the correct answer is 144.
