Question

determine the value(s) for which the rational expression \frac{8q + 8}{99q^2 - 74q - 40} is undefined. if theres more than one value, list them separated by a comma, e.g. q = 2, 3.

provide your answer below:

q =

Explanation:

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<reasoning>

Set the denominator to zero

To find where a rational expression is undefined, we identify values that make the denominator equal to zero.
Using the Undefined Rational Expressions knowledge point:
\[
99q^2 - 74q - 40 = 0
\]

Factor the quadratic trinomial

We factor the quadratic equation using the AC Method and Trinomial Factorization.
We need two numbers that multiply to \(a \cdot c = 99 \cdot (-40) = -3960\) and add to \(b = -74\).
Let's find factors of \(-3960\) that differ by \(-74\):

• \(-110 \cdot 36 = -3960\)
• \(-110 + 36 = -74\)

Rewrite the middle term and factor by grouping:
\[

$$\begin{aligned} 99q^2 - 110q + 36q - 40 &= 0 \\ 11q(9q - 10) + 4(9q - 10) &= 0 \\ (11q + 4)(9q - 10) &= 0 \end{aligned}$$

\]

Solve for q

Using the Zero Product Property to find the roots:
\[

$$\begin{aligned} 11q + 4 = 0 &\implies q = -\frac{4}{11} \\ 9q - 10 = 0 &\implies q = \frac{10}{9} \end{aligned}$$

\]
</reasoning>

<answer>
Determine the value(s) for which the rational expression \(\frac{8q + 8}{99q^2 - 74q - 40}\) is undefined. If there's more than one value, list them separated by a comma, e.g. \(q = 2, 3\).

Provide your answer below:

\(q =\) <blank>\(-\frac{4}{11}, \frac{10}{9}\)</blank>
</answer>

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Answer:

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<reasoning>

Set the denominator to zero

To find where a rational expression is undefined, we identify values that make the denominator equal to zero.
Using the Undefined Rational Expressions knowledge point:
\[
99q^2 - 74q - 40 = 0
\]

Factor the quadratic trinomial

We factor the quadratic equation using the AC Method and Trinomial Factorization.
We need two numbers that multiply to \(a \cdot c = 99 \cdot (-40) = -3960\) and add to \(b = -74\).
Let's find factors of \(-3960\) that differ by \(-74\):

• \(-110 \cdot 36 = -3960\)
• \(-110 + 36 = -74\)

Rewrite the middle term and factor by grouping:
\[

$$\begin{aligned} 99q^2 - 110q + 36q - 40 &= 0 \\ 11q(9q - 10) + 4(9q - 10) &= 0 \\ (11q + 4)(9q - 10) &= 0 \end{aligned}$$

\]

Solve for q

Using the Zero Product Property to find the roots:
\[

$$\begin{aligned} 11q + 4 = 0 &\implies q = -\frac{4}{11} \\ 9q - 10 = 0 &\implies q = \frac{10}{9} \end{aligned}$$

\]
</reasoning>

<answer>
Determine the value(s) for which the rational expression \(\frac{8q + 8}{99q^2 - 74q - 40}\) is undefined. If there's more than one value, list them separated by a comma, e.g. \(q = 2, 3\).

Provide your answer below:

\(q =\) <blank>\(-\frac{4}{11}, \frac{10}{9}\)</blank>
</answer>

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