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Squaring both sides gives \(\displaystyle x^{2} + 10 x + 25 = 9 x + 115\). The equation is quadratic setting it equal to zero gives \(\displaystyle x^{2} + x - 90 = 0\). Factoring gives \(\displaystyle (x - 9)(x + 10)=0\) so the possible solutions are \(\displaystyle x = 9\) and \(\displaystyle x = -10\). Checking the solution \(\displaystyle x = 9\) in the original equation gives \(\displaystyle 14 = 14\). The solution checks, so \(\displaystyle x = 9\) is a true solution. Checking the solution \(\displaystyle x = -10\) in the original equation gives \(\displaystyle -5 = 5\). The solution does no check, so \(\displaystyle x = -10\) is an extraneous solution.

Download \(\LaTeX\)

\begin{question}Solve $x + 5 = \sqrt{9 x + 115}$.
    \soln{10cm}{Squaring both sides gives $x^{2} + 10 x + 25 = 9 x + 115$. The equation is quadratic setting it equal to zero gives $x^{2} + x - 90 = 0$. Factoring gives $(x - 9)(x + 10)=0$ so the possible solutions are $x = 9$ and $x = -10$. Checking the solution $x = 9$ in the original equation gives $14 = 14$. The solution checks, so $x = 9$ is a true solution. Checking the solution $x = -10$ in the original equation gives $-5 = 5$. The solution does no check, so $x = -10$ is an extraneous solution. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

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\newcounter{questioncount}
\stepcounter{questioncount}

\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}

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\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
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\newcommand{\hideifShowSolution}[1]{%
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\everymath{\displaystyle}
\ShowSolutiontrue

\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Solve  <img class="equation_image" title=" \displaystyle x + 5 = \sqrt{9 x + 115} " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%205%20%3D%20%5Csqrt%7B9%20x%20%2B%20115%7D%20" alt="LaTeX:  \displaystyle x + 5 = \sqrt{9 x + 115} " data-equation-content=" \displaystyle x + 5 = \sqrt{9 x + 115} " /> .</p> </p>

HTML for Canvas

<p> <p>Squaring both sides gives  <img class="equation_image" title=" \displaystyle x^{2} + 10 x + 25 = 9 x + 115 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%2010%20x%20%2B%2025%20%3D%209%20x%20%2B%20115%20" alt="LaTeX:  \displaystyle x^{2} + 10 x + 25 = 9 x + 115 " data-equation-content=" \displaystyle x^{2} + 10 x + 25 = 9 x + 115 " /> . The equation is quadratic setting it equal to zero gives  <img class="equation_image" title=" \displaystyle x^{2} + x - 90 = 0 " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20%2B%20x%20-%2090%20%3D%200%20" alt="LaTeX:  \displaystyle x^{2} + x - 90 = 0 " data-equation-content=" \displaystyle x^{2} + x - 90 = 0 " /> . Factoring gives  <img class="equation_image" title=" \displaystyle (x - 9)(x + 10)=0 " src="/equation_images/%20%5Cdisplaystyle%20%28x%20-%209%29%28x%20%2B%2010%29%3D0%20" alt="LaTeX:  \displaystyle (x - 9)(x + 10)=0 " data-equation-content=" \displaystyle (x - 9)(x + 10)=0 " />  so the possible solutions are  <img class="equation_image" title=" \displaystyle x = 9 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%209%20" alt="LaTeX:  \displaystyle x = 9 " data-equation-content=" \displaystyle x = 9 " />  and  <img class="equation_image" title=" \displaystyle x = -10 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-10%20" alt="LaTeX:  \displaystyle x = -10 " data-equation-content=" \displaystyle x = -10 " /> . Checking the solution  <img class="equation_image" title=" \displaystyle x = 9 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%209%20" alt="LaTeX:  \displaystyle x = 9 " data-equation-content=" \displaystyle x = 9 " />  in the original equation gives  <img class="equation_image" title=" \displaystyle 14 = 14 " src="/equation_images/%20%5Cdisplaystyle%2014%20%3D%2014%20" alt="LaTeX:  \displaystyle 14 = 14 " data-equation-content=" \displaystyle 14 = 14 " /> . The solution checks, so  <img class="equation_image" title=" \displaystyle x = 9 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%209%20" alt="LaTeX:  \displaystyle x = 9 " data-equation-content=" \displaystyle x = 9 " />  is a true solution. Checking the solution  <img class="equation_image" title=" \displaystyle x = -10 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-10%20" alt="LaTeX:  \displaystyle x = -10 " data-equation-content=" \displaystyle x = -10 " />  in the original equation gives  <img class="equation_image" title=" \displaystyle -5 = 5 " src="/equation_images/%20%5Cdisplaystyle%20-5%20%3D%205%20" alt="LaTeX:  \displaystyle -5 = 5 " data-equation-content=" \displaystyle -5 = 5 " /> . The solution does no check, so  <img class="equation_image" title=" \displaystyle x = -10 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-10%20" alt="LaTeX:  \displaystyle x = -10 " data-equation-content=" \displaystyle x = -10 " />  is an extraneous solution. </p> </p>