To solve the expression [tex]\(\frac{(4c + 9d)^2}{c^2 - 3d^2}\)[/tex] given [tex]\(c = 3\)[/tex] and [tex]\(d = -2\)[/tex], let's break it down step-by-step:
1. Substitute the values of [tex]\(c\)[/tex] and [tex]\(d\)[/tex] into the expression:
[tex]\[
c = 3, \quad d = -2
\][/tex]
2. Calculate the numerator [tex]\( (4c + 9d)^2 \)[/tex]:
[tex]\[
4c + 9d = 4(3) + 9(-2)
\][/tex]
[tex]\[
= 12 - 18
\][/tex]
[tex]\[
= -6
\][/tex]
Now, square the result:
[tex]\[
(-6)^2 = 36
\][/tex]
So, the numerator is 36.
3. Calculate the denominator [tex]\( c^2 - 3d^2 \)[/tex]:
[tex]\[
c^2 - 3d^2 = (3)^2 - 3(-2)^2
\][/tex]
[tex]\[
= 9 - 3(4)
\][/tex]
[tex]\[
= 9 - 12
\][/tex]
[tex]\[
= -3
\][/tex]
So, the denominator is -3.
4. Form the fraction using the calculated numerator and denominator:
[tex]\[
\frac{(4c + 9d)^2}{c^2 - 3d^2} = \frac{36}{-3}
\][/tex]
5. Simplify the fraction:
[tex]\[
\frac{36}{-3} = -12
\][/tex]
Therefore, the final value of the expression is:
[tex]\[
\frac{(4c + 9d)^2}{c^2 - 3d^2} = -12
\][/tex]
