What value of [tex]$c[tex]$[/tex] makes [tex]$[/tex]x^2 - 12x + c$[/tex] a perfect square trinomial?

A. [tex]$-36$[/tex]

B. [tex]$-24$[/tex]

C. [tex]$24$[/tex]

D. [tex]$36$[/tex]



Answer :

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]