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Questions: Algebra BusinessCalculus
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Solve. \(\displaystyle \sqrt{x + 16}=\sqrt{x - 8} - 2\)
Squaring both sides gives \(\displaystyle x + 16=x - 4 \sqrt{x - 8} - 4\). Isolating the radical gives \(\displaystyle -5=\sqrt{x - 8}\) Squaring again gives \(\displaystyle 25=x - 8\). Solving for \(\displaystyle x\) gives \(\displaystyle x=33\). Checking the solution, \(\displaystyle x = 33\), in the original equation gives \(\displaystyle 7 = 3\) which is false so it is an extraneous solution and there is no solution.
\begin{question}Solve. $\sqrt{x + 16}=\sqrt{x - 8} - 2$
\soln{10cm}{Squaring both sides gives $x + 16=x - 4 \sqrt{x - 8} - 4$. Isolating the radical gives $-5=\sqrt{x - 8}$ Squaring again gives $25=x - 8$. Solving for $x$ gives $x=33$. Checking the solution, $x = 33$, in the original equation gives $7 = 3$ which is false so it is an extraneous solution and there is no solution.}
\end{question}
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\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Solve. <img class="equation_image" title=" \displaystyle \sqrt{x + 16}=\sqrt{x - 8} - 2 " src="/equation_images/%20%5Cdisplaystyle%20%5Csqrt%7Bx%20%2B%2016%7D%3D%5Csqrt%7Bx%20-%208%7D%20-%202%20" alt="LaTeX: \displaystyle \sqrt{x + 16}=\sqrt{x - 8} - 2 " data-equation-content=" \displaystyle \sqrt{x + 16}=\sqrt{x - 8} - 2 " /> </p> </p><p> <p>Squaring both sides gives <img class="equation_image" title=" \displaystyle x + 16=x - 4 \sqrt{x - 8} - 4 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2016%3Dx%20-%204%20%5Csqrt%7Bx%20-%208%7D%20-%204%20" alt="LaTeX: \displaystyle x + 16=x - 4 \sqrt{x - 8} - 4 " data-equation-content=" \displaystyle x + 16=x - 4 \sqrt{x - 8} - 4 " /> . Isolating the radical gives <img class="equation_image" title=" \displaystyle -5=\sqrt{x - 8} " src="/equation_images/%20%5Cdisplaystyle%20-5%3D%5Csqrt%7Bx%20-%208%7D%20" alt="LaTeX: \displaystyle -5=\sqrt{x - 8} " data-equation-content=" \displaystyle -5=\sqrt{x - 8} " /> Squaring again gives <img class="equation_image" title=" \displaystyle 25=x - 8 " src="/equation_images/%20%5Cdisplaystyle%2025%3Dx%20-%208%20" alt="LaTeX: \displaystyle 25=x - 8 " data-equation-content=" \displaystyle 25=x - 8 " /> . Solving for <img class="equation_image" title=" \displaystyle x " src="/equation_images/%20%5Cdisplaystyle%20x%20" alt="LaTeX: \displaystyle x " data-equation-content=" \displaystyle x " /> gives <img class="equation_image" title=" \displaystyle x=33 " src="/equation_images/%20%5Cdisplaystyle%20x%3D33%20" alt="LaTeX: \displaystyle x=33 " data-equation-content=" \displaystyle x=33 " /> . Checking the solution, <img class="equation_image" title=" \displaystyle x = 33 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%2033%20" alt="LaTeX: \displaystyle x = 33 " data-equation-content=" \displaystyle x = 33 " /> , in the original equation gives <img class="equation_image" title=" \displaystyle 7 = 3 " src="/equation_images/%20%5Cdisplaystyle%207%20%3D%203%20" alt="LaTeX: \displaystyle 7 = 3 " data-equation-content=" \displaystyle 7 = 3 " /> which is false so it is an extraneous solution and there is no solution.</p> </p>