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Algebra
Radicals
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Solve. \(\displaystyle \sqrt{x + 20}=\sqrt{x - 6} - 4\)

Squaring both sides gives \(\displaystyle x + 20=x - 8 \sqrt{x - 6} + 10\). Isolating the radical gives \(\displaystyle - \frac{5}{4}=\sqrt{x - 6}\) Squaring again gives \(\displaystyle \frac{25}{16}=x - 6\). Solving for \(\displaystyle x\) gives \(\displaystyle x=\frac{121}{16}\). Checking the solution, \(\displaystyle x = \frac{121}{16}\), in the original equation gives \(\displaystyle \frac{21}{4} = - \frac{11}{4}\) which is false so it is an extraneous solution and there is no solution.

Download \(\LaTeX\) 

\begin{question}Solve. $\sqrt{x + 20}=\sqrt{x - 6} - 4$
    \soln{10cm}{Squaring both sides gives $x + 20=x - 8 \sqrt{x - 6} + 10$. Isolating the radical gives $- \frac{5}{4}=\sqrt{x - 6}$ Squaring again gives $\frac{25}{16}=x - 6$.  Solving for $x$ gives $x=\frac{121}{16}$. Checking the solution, $x = \frac{121}{16}$, in the original equation gives $\frac{21}{4} = - \frac{11}{4}$ 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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HTML for Canvas 
<p> <p>Solve.  <img class="equation_image" title=" \displaystyle \sqrt{x + 20}=\sqrt{x - 6} - 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Csqrt%7Bx%20%2B%2020%7D%3D%5Csqrt%7Bx%20-%206%7D%20-%204%20" alt="LaTeX:  \displaystyle \sqrt{x + 20}=\sqrt{x - 6} - 4 " data-equation-content=" \displaystyle \sqrt{x + 20}=\sqrt{x - 6} - 4 " /> </p> </p>
HTML for Canvas
<p> <p>Squaring both sides gives  <img class="equation_image" title=" \displaystyle x + 20=x - 8 \sqrt{x - 6} + 10 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2020%3Dx%20-%208%20%5Csqrt%7Bx%20-%206%7D%20%2B%2010%20" alt="LaTeX:  \displaystyle x + 20=x - 8 \sqrt{x - 6} + 10 " data-equation-content=" \displaystyle x + 20=x - 8 \sqrt{x - 6} + 10 " /> . Isolating the radical gives  <img class="equation_image" title=" \displaystyle - \frac{5}{4}=\sqrt{x - 6} " src="/equation_images/%20%5Cdisplaystyle%20-%20%5Cfrac%7B5%7D%7B4%7D%3D%5Csqrt%7Bx%20-%206%7D%20" alt="LaTeX:  \displaystyle - \frac{5}{4}=\sqrt{x - 6} " data-equation-content=" \displaystyle - \frac{5}{4}=\sqrt{x - 6} " />  Squaring again gives  <img class="equation_image" title=" \displaystyle \frac{25}{16}=x - 6 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B25%7D%7B16%7D%3Dx%20-%206%20" alt="LaTeX:  \displaystyle \frac{25}{16}=x - 6 " data-equation-content=" \displaystyle \frac{25}{16}=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{121}{16} " src="/equation_images/%20%5Cdisplaystyle%20x%3D%5Cfrac%7B121%7D%7B16%7D%20" alt="LaTeX:  \displaystyle x=\frac{121}{16} " data-equation-content=" \displaystyle x=\frac{121}{16} " /> . Checking the solution,  <img class="equation_image" title=" \displaystyle x = \frac{121}{16} " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20%5Cfrac%7B121%7D%7B16%7D%20" alt="LaTeX:  \displaystyle x = \frac{121}{16} " data-equation-content=" \displaystyle x = \frac{121}{16} " /> , in the original equation gives  <img class="equation_image" title=" \displaystyle \frac{21}{4} = - \frac{11}{4} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B21%7D%7B4%7D%20%3D%20-%20%5Cfrac%7B11%7D%7B4%7D%20" alt="LaTeX:  \displaystyle \frac{21}{4} = - \frac{11}{4} " data-equation-content=" \displaystyle \frac{21}{4} = - \frac{11}{4} " />  which is false so it is an extraneous solution and there is no solution.</p> </p>