Bond Carrillo
04/16/2024 · Senior High School
Solve the equation, eliminating any extraneous solutions. \( \frac{x^{2}+5}{(x+5)}=\frac{30}{(x+5)} \)
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Step-by-step Solution
Solve the equation \( \frac{x^{2}+5}{(x+5)}=\frac{30}{(x+5)} \).
Solve the equation by following steps:
- step0: Solve for \(x\):
\(\frac{x^{2}+5}{\left(x+5\right)}=\frac{30}{\left(x+5\right)}\)
- step1: Find the domain:
\(\frac{x^{2}+5}{\left(x+5\right)}=\frac{30}{\left(x+5\right)},x\neq -5\)
- step2: Remove the parentheses:
\(\frac{x^{2}+5}{x+5}=\frac{30}{x+5}\)
- step3: Cross multiply:
\(\left(x^{2}+5\right)\left(x+5\right)=\left(x+5\right)\times 30\)
- step4: Simplify the equation:
\(\left(x^{2}+5\right)\left(x+5\right)=30\left(x+5\right)\)
- step5: Calculate:
\(x^{3}+5x^{2}+5x+25=30\left(x+5\right)\)
- step6: Calculate:
\(x^{3}+5x^{2}+5x+25=30x+150\)
- step7: Move the expression to the left side:
\(x^{3}+5x^{2}+5x+25-\left(30x+150\right)=0\)
- step8: Calculate:
\(x^{3}+5x^{2}-25x-125=0\)
- step9: Factor the expression:
\(\left(x-5\right)\left(x+5\right)^{2}=0\)
- step10: Separate into possible cases:
\(\begin{align}&x-5=0\\&\left(x+5\right)^{2}=0\end{align}\)
- step11: Solve the equation:
\(\begin{align}&x=5\\&x=-5\end{align}\)
- step12: Check if the solution is in the defined range:
\(\begin{align}&x=5\\&x=-5\end{align},x\neq -5\)
- step13: Find the intersection:
\(x=5\)
The solution to the equation \( \frac{x^{2}+5}{(x+5)}=\frac{30}{(x+5)} \) is \( x=5 \).
Quick Answer
The solution to the equation is \( x=5 \).
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