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What value of [tex]$c[tex]$[/tex] makes [tex]$[/tex]x^2 - 24x + c$[/tex] a perfect square trinomial?

A. [tex]$-144$[/tex]
B. [tex]$-48$[/tex]
C. 48
D. 144

Sagot :

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.