# 9.6 Square root equations with applications

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: be able to recognize square root equations and extraneous solutions, be able to sove square root equations.

## Overview

• Square Root Equations And Extraneous Solutions
• Method For Solving Square Root Equations

## Square root equation

A square root equation is an equation that contains a variable under a square root sign. The fact that $\sqrt{x}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\sqrt{x}={\left(\sqrt{x}\right)}^{2}=x$ suggests that we can solve a square root equation by squaring both sides of the equation.

## Extraneous solutions

Squaring both sides of an equation can, however, introduce extraneous solutions. Consider the equation

$x=-6$

The solution is $-6.$ Square both sides.

${x}^{2}={\left(-6\right)}^{2}$
${x}^{2}=36$

This equation has two solutions, $-6$ and $+6.$ The $+6$ is an extraneous solution since it does not check in the original equation: $+6\ne -6.$

## Solving square root equations

1. Isolate a radical. This means get a square root expression by itself on one side of the equal sign.
2. Square both sides of the equation.
3. Simplify the equation by combining like terms.
4. Repeat step 1 if radicals are still present.
5. Obtain potential solutions by solving the resulting non-square root equation.
6. Check each potential solution by substitution into the original equation.

## Sample set a

Solve each square root equation.

$\begin{array}{cccccc}\hfill & \hfill \sqrt{x}& =\hfill & 8.\hfill & \hfill & \begin{array}{l}\text{The radical is isolated Square both sides}\text{.}\hfill \end{array}\hfill \\ \hfill & {\left(\sqrt{x}\right)}^{2}\hfill & =\hfill & {8}^{2}\hfill & \hfill & \hfill \\ \hfill & \hfill x& =\hfill & 64\hfill & \hfill & \text{Check this potential solution}\text{.}\hfill \\ Check:\hfill & \hfill \sqrt{64}& =\hfill & 8\hfill & \hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill 8& =\hfill & 8\hfill & \hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct.}\hfill \\ \text{64 is the solution}\text{.}\hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$

$\begin{array}{cccccc}\hfill & \hfill \sqrt{y-3}& =\hfill & 4.\hfill & \hfill & \begin{array}{l}\text{The radical is isolated. Square both sides}\text{.}\hfill \end{array}\hfill \\ \hfill & \hfill y-3& =\hfill & 16\hfill & \hfill & \text{Solve this nonradical equation}\text{.}\hfill \\ \hfill & \hfill y& =\hfill & 19\hfill & \hfill & \text{Check this potential solution}\text{.}\hfill \\ Check:\hfill & \sqrt{19-3}\hfill & =\hfill & \sqrt{16}\hfill & \hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill \sqrt{16}& =\hfill & 4\hfill & \hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill 4& =\hfill & 4\hfill & \hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct.}\hfill \\ \text{19 is the solution}\text{.}\hfill & \hfill & \hfill & \hfill & \hfill & \hfill \end{array}$

$\begin{array}{ccccccc}\hfill & \hfill & \hfill \sqrt{2m+3}-\sqrt{m-8}& =\hfill & 0.\hfill & \hfill & \text{Isolate either radical}\text{.}\hfill \\ \hfill & \hfill & \hfill \sqrt{2m+3}& =\hfill & \sqrt{m-8}\hfill & \hfill & \text{Square both sides}\text{.}\hfill \\ \hfill & \hfill & \hfill 2m+3& =\hfill & m-8\hfill & \hfill & \text{Solve this nonradical equation}\text{.}\hfill \\ \hfill & \hfill & \hfill m& =\hfill & -11\hfill & \hfill & \text{Check this potential solution}\text{.}\hfill \\ Check:\hfill & \hfill & \hfill \sqrt{2\left(-11\right)+3}-\sqrt{\left(-11\right)-8}& =\hfill & 0\hfill & \hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill \sqrt{-22+3}-\sqrt{-19}& =\hfill & 0\hfill & \hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \end{array}$
$\text{Since\hspace{0.17em}}\sqrt{-19}\text{\hspace{0.17em}is not a real number, the potential solution of\hspace{0.17em}}m=-11\text{\hspace{0.17em}does not check}\text{. This equation has no real solution}\text{.}$

$\begin{array}{ccc}\sqrt{4x-5}=-6.& & \text{By\hspace{0.17em}inspection,\hspace{0.17em}this\hspace{0.17em}equation\hspace{0.17em}has\hspace{0.17em}no\hspace{0.17em}real\hspace{0.17em}solution}\text{.}\end{array}$
The symbol, $\sqrt{\begin{array}{c}\\ \end{array}}$ , signifies the positive square root and not the negative square root.

## Practice set a

Solve each square root equation.

$\sqrt{y}=14$

$y=196$

$\sqrt{a-7}=5$

$a=32$

$\sqrt{3a+8}-\sqrt{2a+5}=0$

$a=-3$ is extraneous, no real solution

$\sqrt{m-4}=-11$

no real solution

## Exercises

For the following problems, solve the square root equations.

$\sqrt{x}=5$

$x=25$

$\sqrt{y}=7$

$\sqrt{a}=10$

$a=100$

$\sqrt{c}=12$

$\sqrt{x}=-3$

no solution

$\sqrt{y}=-6$

$\sqrt{x}=0$

$x=0$

$\sqrt{x}=1$

$\sqrt{x+3}=3$

$x=6$

$\sqrt{y-5}=5$

$\sqrt{a+2}=6$

$a=34$

$\sqrt{y+7}=9$

$\sqrt{y-4}-4=0$

$y=20$

$\sqrt{x-10}-10=0$

$\sqrt{x-16}=0$

$x=16$

$\sqrt{y-25}=0$

$\sqrt{6m-4}=\sqrt{5m-1}$

$m=3$

$\sqrt{5x+6}=\sqrt{3x+7}$

$\sqrt{7a+6}=\sqrt{3a-18}$

no solution

$\sqrt{4x+3}=\sqrt{x-9}$

$\sqrt{10a-7}-\sqrt{2a+9}=0$

$a=2$

$\sqrt{12k-5}-\sqrt{9k+10}=0$

$\sqrt{x-6}-\sqrt{3x-8}=0$

no solution

$\sqrt{4a-5}-\sqrt{7a-20}=0$

$\sqrt{2m-6}=\sqrt{m-2}$

$m=4$

$\sqrt{6r-11}=\sqrt{5r+3}$

$\sqrt{3x+1}=\sqrt{2x-6}$

no solution

$\sqrt{x-7}-\sqrt{5x+1}=0$

$\sqrt{2a+9}-\sqrt{a-4}=0$

no solution

At a certain electronics company, the daily output $Q$ is related to the number of people $A$ on the assembly line by $Q=400+10\sqrt{A+125}.$
(a) Determine the daily output if there are 44 people on the assembly line.

(b) Determine how many people are needed on the assembly line if the daily output is to be 520.

At a store, the daily number of sales $S$ is approximately related to the number of employees $E$ by $S=100+15\sqrt{E+6}$
(a) Determine the approximate number of sales if there are 19 employees.

(b) Determine the number of employees the store would need to produce 310 sales.

$\left(a\right)S=175;\text{\hspace{0.17em}}\left(b\right)E=190$

The resonance frequency $f$ in an electronic circuit containing inductance $L$ and capacitance $C$ in series is given by

$f=\frac{1}{2\pi \sqrt{LC}}$

(a) Determine the resonance frequency in an electronic circuit if the inductance is 4 and the capacitance is $0.0001$ . Use $\pi =3.14.$

(b) Determine the inductance in an electric circuit if the resonance frequency is $7.12$ and the capacitance is $0.0001$ . Use $\pi =3.14.$

If two magnetic poles of strength $m$ and ${m}^{\text{'}}$ units are at a distance $r$ centimeters (cm) apart, the force $F$ of repulsion in air between them is given by

$F=\frac{m{m}^{\text{'}}}{{r}^{2}}$

(a) Determine the force of repulsion if two magnetic poles of strengths 20 and 40 units are 5 cm apart in air.

(b) Determine how far apart are two magnetic poles of strengths 30 and 40 units if the force of repulsion in air between them is $0.0001$ .

$\left(a\right)F=32\text{\hspace{0.17em}}\left(b\right)r=8\text{cm}$

The velocity $V$ in feet per second of outflow of a liquid from an orifice is given by $V=8\sqrt{h},$ where $h$ is the height in feet of the liquid above the opening.
(a) Determine the velocity of outflow of a liquid from an orifice that is 9 feet below the top surface of a liquid ( $V$ is in feet/sec).

(b) Determine how high a liquid is above an orifice if the velocity of outflow is 81 feet/second.

The period $T$ in seconds of a simple pendulum of length $L$ in feet is given by $T=2\pi \sqrt{\frac{L}{32}}.$

(a) Determine the period of a simple pendulum that is 2 feet long. Use $\pi =3.14.$

(b) Determine the length in feet of a simple pendulum whose period is $10.8772$ seconds. Use $\pi =3.14.$

$\left(a\right)T=1.57\text{\hspace{0.17em}sec\hspace{0.17em}}\left(b\right)L=95.99\text{\hspace{0.17em}cm}$

The kinetic energy $KE$ in foot pounds of a body of mass $m$ in slugs moving with a velocity $v$ in feet/sec is given by

$KE=\frac{1}{2}m{v}^{2}$

(a) Determine the kinetic energy of a 2-slug body moving with a velocity of 4 ft/sec.

(b) Determine the velocity in feet/sec of a 4-slug body if its kinetic energy is 50 foot pounds.

## Exercises for review

( [link] ) Write $\frac{{x}^{10}{y}^{3}{\left(x+7\right)}^{4}}{{x}^{-2}{y}^{3}{\left(x+7\right)}^{-1}}$ so that only positive exponents appear.

${x}^{12}{\left(x+7\right)}^{5}$

( [link] ) Classify $x+4=x+7$ as an identity, a contradiction, or a conditional equation.

( [link] ) Supply the missing words. In the coordinate plane, lines with slope rise and lines with slope fall.

positive; negative

( [link] ) Simplify $\sqrt{{\left(x+3\right)}^{4}{\left(x-2\right)}^{6}}.$

( [link] ) Simplify $\left(3+\sqrt{5}\right)\left(4-\sqrt{5}\right).$

$7+\sqrt{5}$

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