Question
X. 6 Add and subtract rational expressions FEX Simplify. Express your answer as a single fraction in simplest form. \( \frac{9 t}{9 t^{2}+14 t+5}-\frac{t}{9 t+5} \)
Ask by Robbins Williams. in the United States
Jan 30,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( \frac{8t - t^{2}}{9t^{2} + 14t + 5} \).
Solution
Calculate or simplify the expression \( \frac{9t}{9t^{2}+14t+5}-\frac{t}{9t+5} \).
Simplify the expression by following steps:
- step0: Solution:
\(\frac{9t}{9t^{2}+14t+5}-\frac{t}{9t+5}\)
- step1: Factor the expression:
\(\frac{9t}{\left(t+1\right)\left(9t+5\right)}-\frac{t}{9t+5}\)
- step2: Reduce fractions to a common denominator:
\(\frac{9t}{\left(t+1\right)\left(9t+5\right)}-\frac{t\left(t+1\right)}{\left(9t+5\right)\left(t+1\right)}\)
- step3: Rewrite the expression:
\(\frac{9t}{\left(t+1\right)\left(9t+5\right)}-\frac{t\left(t+1\right)}{\left(t+1\right)\left(9t+5\right)}\)
- step4: Transform the expression:
\(\frac{9t-t\left(t+1\right)}{\left(t+1\right)\left(9t+5\right)}\)
- step5: Multiply the terms:
\(\frac{9t-\left(t^{2}+t\right)}{\left(t+1\right)\left(9t+5\right)}\)
- step6: Subtract the terms:
\(\frac{8t-t^{2}}{\left(t+1\right)\left(9t+5\right)}\)
- step7: Multiply the terms:
\(\frac{8t-t^{2}}{9t^{2}+14t+5}\)
The simplified form of the expression \( \frac{9t}{9t^{2}+14t+5}-\frac{t}{9t+5} \) is \( \frac{8t-t^{2}}{9t^{2}+14t+5} \).
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Simplify this solution Beyond the Answer
To simplify the expression \( \frac{9 t}{9 t^{2}+14 t+5}-\frac{t}{9 t+5} \), first, we need to find a common denominator. The first denominator factors as \( (9t + 5)(t + 1) \). Now, rewrite the fractions with the common denominator: \[ \frac{9 t(9 t + 5)}{(9 t + 5)(9 t + 5)} - \frac{t(9 t + 1)}{(9 t + 5)(9 t + 1)} \] Now, combine the fractions: \[ \frac{9 t(9 t + 5) - t(9 t + 1)}{(9 t + 5)(9 t + 1)} \] Simplify the numerator step by step: \[ = \frac{81 t^2 + 45 t - 9 t^2 - t}{(9 t + 5)(9 t + 1)} = \frac{72 t^2 + 44 t}{(9 t + 5)(9 t + 1)} \] Finally, factor out the common term in the numerator: \[ = \frac{4t(18 t + 11)}{(9 t + 5)(9 t + 1)} \] This is the simplified form.
