Evaluate the following expression for [tex]\( c = 3 \)[/tex] and [tex]\( d = -2 \)[/tex]:

[tex]\[ \frac{(4c + 9d)^2}{c^2 - 3d^2} \][/tex]



Answer :

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]