To determine the nonpermissible values for [tex]\( a \)[/tex] in the expression [tex]\(\frac{a^2}{-3a-6}\)[/tex], we need to find the values of [tex]\( a \)[/tex] that make the denominator equal to zero. A fraction is undefined when its denominator is zero, so we need to avoid these values to ensure the expression is defined.
Let's set the denominator equal to zero and solve for [tex]\( a \)[/tex]:
[tex]\[
-3a - 6 = 0
\][/tex]
To solve this equation, follow these steps:
1. Add 6 to both sides:
[tex]\[
-3a - 6 + 6 = 0 + 6
\][/tex]
Simplifying, we get:
[tex]\[
-3a = 6
\][/tex]
2. Divide both sides by -3:
[tex]\[
\frac{-3a}{-3} = \frac{6}{-3}
\][/tex]
Simplifying, we get:
[tex]\[
a = -2
\][/tex]
Thus, the value of [tex]\( a \)[/tex] that makes the denominator zero is [tex]\( -2 \)[/tex]. This is the nonpermissible value because [tex]\( \frac{a^2}{-3a-6} \)[/tex] is undefined when [tex]\( a = -2 \)[/tex].
Conclusion: The nonpermissible replacement for [tex]\( a \)[/tex] in the expression [tex]\(\frac{a^2}{-3a-6}\)[/tex] is [tex]\( -2 \)[/tex].
