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Algebra
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Solve. \(\displaystyle \sqrt{x + 26}=\sqrt{x - 6} - 3\)

Squaring both sides gives \(\displaystyle x + 26=x - 6 \sqrt{x - 6} + 3\). Isolating the radical gives \(\displaystyle - \frac{23}{6}=\sqrt{x - 6}\) Squaring again gives \(\displaystyle \frac{529}{36}=x - 6\). Solving for \(\displaystyle x\) gives \(\displaystyle x=\frac{745}{36}\). Checking the solution, \(\displaystyle x = \frac{745}{36}\), in the original equation gives \(\displaystyle \frac{41}{6} = \frac{5}{6}\) which is false so it is an extraneous solution and there is no solution.

Download \(\LaTeX\) 

\begin{question}Solve. $\sqrt{x + 26}=\sqrt{x - 6} - 3$
    \soln{10cm}{Squaring both sides gives $x + 26=x - 6 \sqrt{x - 6} + 3$. Isolating the radical gives $- \frac{23}{6}=\sqrt{x - 6}$ Squaring again gives $\frac{529}{36}=x - 6$.  Solving for $x$ gives $x=\frac{745}{36}$. Checking the solution, $x = \frac{745}{36}$, in the original equation gives $\frac{41}{6} = \frac{5}{6}$ which is false so it is an extraneous solution and there is no solution.}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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    \soln{9cm}{The solution goes here.}

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HTML for Canvas 
<p> <p>Solve.  <img class="equation_image" title=" \displaystyle \sqrt{x + 26}=\sqrt{x - 6} - 3 " src="/equation_images/%20%5Cdisplaystyle%20%5Csqrt%7Bx%20%2B%2026%7D%3D%5Csqrt%7Bx%20-%206%7D%20-%203%20" alt="LaTeX:  \displaystyle \sqrt{x + 26}=\sqrt{x - 6} - 3 " data-equation-content=" \displaystyle \sqrt{x + 26}=\sqrt{x - 6} - 3 " /> </p> </p>
HTML for Canvas
<p> <p>Squaring both sides gives  <img class="equation_image" title=" \displaystyle x + 26=x - 6 \sqrt{x - 6} + 3 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2026%3Dx%20-%206%20%5Csqrt%7Bx%20-%206%7D%20%2B%203%20" alt="LaTeX:  \displaystyle x + 26=x - 6 \sqrt{x - 6} + 3 " data-equation-content=" \displaystyle x + 26=x - 6 \sqrt{x - 6} + 3 " /> . Isolating the radical gives  <img class="equation_image" title=" \displaystyle - \frac{23}{6}=\sqrt{x - 6} " src="/equation_images/%20%5Cdisplaystyle%20-%20%5Cfrac%7B23%7D%7B6%7D%3D%5Csqrt%7Bx%20-%206%7D%20" alt="LaTeX:  \displaystyle - \frac{23}{6}=\sqrt{x - 6} " data-equation-content=" \displaystyle - \frac{23}{6}=\sqrt{x - 6} " />  Squaring again gives  <img class="equation_image" title=" \displaystyle \frac{529}{36}=x - 6 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B529%7D%7B36%7D%3Dx%20-%206%20" alt="LaTeX:  \displaystyle \frac{529}{36}=x - 6 " data-equation-content=" \displaystyle \frac{529}{36}=x - 6 " /> .  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=\frac{745}{36} " src="/equation_images/%20%5Cdisplaystyle%20x%3D%5Cfrac%7B745%7D%7B36%7D%20" alt="LaTeX:  \displaystyle x=\frac{745}{36} " data-equation-content=" \displaystyle x=\frac{745}{36} " /> . Checking the solution,  <img class="equation_image" title=" \displaystyle x = \frac{745}{36} " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20%5Cfrac%7B745%7D%7B36%7D%20" alt="LaTeX:  \displaystyle x = \frac{745}{36} " data-equation-content=" \displaystyle x = \frac{745}{36} " /> , in the original equation gives  <img class="equation_image" title=" \displaystyle \frac{41}{6} = \frac{5}{6} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B41%7D%7B6%7D%20%3D%20%5Cfrac%7B5%7D%7B6%7D%20" alt="LaTeX:  \displaystyle \frac{41}{6} = \frac{5}{6} " data-equation-content=" \displaystyle \frac{41}{6} = \frac{5}{6} " />  which is false so it is an extraneous solution and there is no solution.</p> </p>