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Algebra
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Solve. \(\displaystyle \sqrt{x + 24}=\sqrt{x - 6} - 1\)

Squaring both sides gives \(\displaystyle x + 24=x - 2 \sqrt{x - 6} - 5\). Isolating the radical gives \(\displaystyle - \frac{29}{2}=\sqrt{x - 6}\) Squaring again gives \(\displaystyle \frac{841}{4}=x - 6\). Solving for \(\displaystyle x\) gives \(\displaystyle x=\frac{865}{4}\). Checking the solution, \(\displaystyle x = \frac{865}{4}\), in the original equation gives \(\displaystyle \frac{31}{2} = \frac{27}{2}\) which is false so it is an extraneous solution and there is no solution.

Download \(\LaTeX\) 

\begin{question}Solve. $\sqrt{x + 24}=\sqrt{x - 6} - 1$
    \soln{10cm}{Squaring both sides gives $x + 24=x - 2 \sqrt{x - 6} - 5$. Isolating the radical gives $- \frac{29}{2}=\sqrt{x - 6}$ Squaring again gives $\frac{841}{4}=x - 6$.  Solving for $x$ gives $x=\frac{865}{4}$. Checking the solution, $x = \frac{865}{4}$, in the original equation gives $\frac{31}{2} = \frac{27}{2}$ 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 + 24}=\sqrt{x - 6} - 1 " src="/equation_images/%20%5Cdisplaystyle%20%5Csqrt%7Bx%20%2B%2024%7D%3D%5Csqrt%7Bx%20-%206%7D%20-%201%20" alt="LaTeX:  \displaystyle \sqrt{x + 24}=\sqrt{x - 6} - 1 " data-equation-content=" \displaystyle \sqrt{x + 24}=\sqrt{x - 6} - 1 " /> </p> </p>
HTML for Canvas
<p> <p>Squaring both sides gives  <img class="equation_image" title=" \displaystyle x + 24=x - 2 \sqrt{x - 6} - 5 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2024%3Dx%20-%202%20%5Csqrt%7Bx%20-%206%7D%20-%205%20" alt="LaTeX:  \displaystyle x + 24=x - 2 \sqrt{x - 6} - 5 " data-equation-content=" \displaystyle x + 24=x - 2 \sqrt{x - 6} - 5 " /> . Isolating the radical gives  <img class="equation_image" title=" \displaystyle - \frac{29}{2}=\sqrt{x - 6} " src="/equation_images/%20%5Cdisplaystyle%20-%20%5Cfrac%7B29%7D%7B2%7D%3D%5Csqrt%7Bx%20-%206%7D%20" alt="LaTeX:  \displaystyle - \frac{29}{2}=\sqrt{x - 6} " data-equation-content=" \displaystyle - \frac{29}{2}=\sqrt{x - 6} " />  Squaring again gives  <img class="equation_image" title=" \displaystyle \frac{841}{4}=x - 6 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B841%7D%7B4%7D%3Dx%20-%206%20" alt="LaTeX:  \displaystyle \frac{841}{4}=x - 6 " data-equation-content=" \displaystyle \frac{841}{4}=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{865}{4} " src="/equation_images/%20%5Cdisplaystyle%20x%3D%5Cfrac%7B865%7D%7B4%7D%20" alt="LaTeX:  \displaystyle x=\frac{865}{4} " data-equation-content=" \displaystyle x=\frac{865}{4} " /> . Checking the solution,  <img class="equation_image" title=" \displaystyle x = \frac{865}{4} " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20%5Cfrac%7B865%7D%7B4%7D%20" alt="LaTeX:  \displaystyle x = \frac{865}{4} " data-equation-content=" \displaystyle x = \frac{865}{4} " /> , in the original equation gives  <img class="equation_image" title=" \displaystyle \frac{31}{2} = \frac{27}{2} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B31%7D%7B2%7D%20%3D%20%5Cfrac%7B27%7D%7B2%7D%20" alt="LaTeX:  \displaystyle \frac{31}{2} = \frac{27}{2} " data-equation-content=" \displaystyle \frac{31}{2} = \frac{27}{2} " />  which is false so it is an extraneous solution and there is no solution.</p> </p>