Curry Schofield
12/13/2023 · High School
Terry and Veron were asked to solve the equation \( \frac{100 x}{(9-5 x)^{2}}=x \)
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Tutor-Verified Answer
Step-by-step Solution
Solve the equation \( \frac{100x}{(9-5x)^2}=x \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{100x}{\left(9-5x\right)^{2}}=x\)
- step1: Find the domain:
\(\frac{100x}{\left(9-5x\right)^{2}}=x ,x\neq \frac{9}{5}\)
- step2: Cross multiply:
\(100x=\left(9-5x\right)^{2}x\)
- step3: Simplify the equation:
\(100x=x\left(9-5x\right)^{2}\)
- step4: Expand the expression:
\(100x=81x-90x^{2}+25x^{3}\)
- step5: Move the expression to the left side:
\(100x-\left(81x-90x^{2}+25x^{3}\right)=0\)
- step6: Subtract the terms:
\(19x+90x^{2}-25x^{3}=0\)
- step7: Factor the expression:
\(x\left(19-5x\right)\left(1+5x\right)=0\)
- step8: Separate into possible cases:
\(\begin{align}&x=0\\&19-5x=0\\&1+5x=0\end{align}\)
- step9: Solve the equation:
\(\begin{align}&x=0\\&x=\frac{19}{5}\\&x=-\frac{1}{5}\end{align}\)
- step10: Check if the solution is in the defined range:
\(\begin{align}&x=0\\&x=\frac{19}{5}\\&x=-\frac{1}{5}\end{align},x\neq \frac{9}{5}\)
- step11: Find the intersection:
\(\begin{align}&x=0\\&x=\frac{19}{5}\\&x=-\frac{1}{5}\end{align}\)
- step12: Rewrite:
\(x_{1}=-\frac{1}{5},x_{2}=0,x_{3}=\frac{19}{5}\)
The solutions to the equation \( \frac{100x}{(9-5x)^{2}}=x \) are \( x = -\frac{1}{5} \), \( x = 0 \), and \( x = \frac{19}{5} \).
Quick Answer
The solutions are \( x = -\frac{1}{5} \), \( x = 0 \), and \( x = \frac{19}{5} \).
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