\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus

Please login to create an exam or a quiz.

Algebra
Radicals
New Random

Solve. \(\displaystyle \sqrt{x + 12}=\sqrt{x + 4} - 3\)

Squaring both sides gives \(\displaystyle x + 12=x - 6 \sqrt{x + 4} + 13\). Isolating the radical gives \(\displaystyle \frac{1}{6}=\sqrt{x + 4}\) Squaring again gives \(\displaystyle \frac{1}{36}=x + 4\). Solving for \(\displaystyle x\) gives \(\displaystyle x=- \frac{143}{36}\). Checking the solution, \(\displaystyle x = - \frac{143}{36}\), in the original equation gives \(\displaystyle \frac{17}{6} = - \frac{17}{6}\) which is false so it is an extraneous solution and there is no solution.

Download \(\LaTeX\) 

\begin{question}Solve. $\sqrt{x + 12}=\sqrt{x + 4} - 3$
    \soln{10cm}{Squaring both sides gives $x + 12=x - 6 \sqrt{x + 4} + 13$. Isolating the radical gives $\frac{1}{6}=\sqrt{x + 4}$ Squaring again gives $\frac{1}{36}=x + 4$.  Solving for $x$ gives $x=- \frac{143}{36}$. Checking the solution, $x = - \frac{143}{36}$, in the original equation gives $\frac{17}{6} = - \frac{17}{6}$ which is false so it is an extraneous solution and there is no solution.}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}

\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}

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

\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\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 \sqrt{x + 12}=\sqrt{x + 4} - 3 " src="/equation_images/%20%5Cdisplaystyle%20%5Csqrt%7Bx%20%2B%2012%7D%3D%5Csqrt%7Bx%20%2B%204%7D%20-%203%20" alt="LaTeX:  \displaystyle \sqrt{x + 12}=\sqrt{x + 4} - 3 " data-equation-content=" \displaystyle \sqrt{x + 12}=\sqrt{x + 4} - 3 " /> </p> </p>
HTML for Canvas
<p> <p>Squaring both sides gives  <img class="equation_image" title=" \displaystyle x + 12=x - 6 \sqrt{x + 4} + 13 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%2012%3Dx%20-%206%20%5Csqrt%7Bx%20%2B%204%7D%20%2B%2013%20" alt="LaTeX:  \displaystyle x + 12=x - 6 \sqrt{x + 4} + 13 " data-equation-content=" \displaystyle x + 12=x - 6 \sqrt{x + 4} + 13 " /> . Isolating the radical gives  <img class="equation_image" title=" \displaystyle \frac{1}{6}=\sqrt{x + 4} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B6%7D%3D%5Csqrt%7Bx%20%2B%204%7D%20" alt="LaTeX:  \displaystyle \frac{1}{6}=\sqrt{x + 4} " data-equation-content=" \displaystyle \frac{1}{6}=\sqrt{x + 4} " />  Squaring again gives  <img class="equation_image" title=" \displaystyle \frac{1}{36}=x + 4 " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B36%7D%3Dx%20%2B%204%20" alt="LaTeX:  \displaystyle \frac{1}{36}=x + 4 " data-equation-content=" \displaystyle \frac{1}{36}=x + 4 " /> .  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{143}{36} " src="/equation_images/%20%5Cdisplaystyle%20x%3D-%20%5Cfrac%7B143%7D%7B36%7D%20" alt="LaTeX:  \displaystyle x=- \frac{143}{36} " data-equation-content=" \displaystyle x=- \frac{143}{36} " /> . Checking the solution,  <img class="equation_image" title=" \displaystyle x = - \frac{143}{36} " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-%20%5Cfrac%7B143%7D%7B36%7D%20" alt="LaTeX:  \displaystyle x = - \frac{143}{36} " data-equation-content=" \displaystyle x = - \frac{143}{36} " /> , in the original equation gives  <img class="equation_image" title=" \displaystyle \frac{17}{6} = - \frac{17}{6} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B17%7D%7B6%7D%20%3D%20-%20%5Cfrac%7B17%7D%7B6%7D%20" alt="LaTeX:  \displaystyle \frac{17}{6} = - \frac{17}{6} " data-equation-content=" \displaystyle \frac{17}{6} = - \frac{17}{6} " />  which is false so it is an extraneous solution and there is no solution.</p> </p>