# 9.7: Square Root Equations with Applications

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

## Square Root Equations And Extraneous Solutions

Square Root Equation

A square root equation is an equation that contains a variable under a square root sign. The fact that $$\sqrt{x} \cdot \sqrt{x} = (\sqrt{x})^2 = x$$ 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.

$$x^2 = (-6)^2$$

$$x^2 = 36$$

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

$$+6 \not = -6$$

## Method For Solving Square Root Equations

##### 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.

##### Example $$\PageIndex{1}$$

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

##### Example $$\PageIndex{2}$$

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

##### Example $$\PageIndex{3}$$

$$\begin{array}{flushleft} & \sqrt{2m + 3} - \sqrt{m - 8} &= 0 & \text{ Isolate either radical }\\ & \sqrt{2m + 3} &= \sqrt{m + 8} & \text{ Square both sides. }\\ & 2m + 3 &= m-8 & \text{ Solve this nonradical equation }\\ & m &= -11 & \text{ Check this potential solution. }\\ \text{Check: } \sqrt{2(-11) + 3} - \sqrt{(-11) - 8} &= 0 & \text{ Is this correct? }\\ & \sqrt{-22 + 3} - \sqrt{-19} &= 0 & \text{ Is this correct? } \end{array}$$

Since $$\sqrt{-19}$$ is not a real number, the potential solution of $$m = -11$$ does not check. This equation has no real solution.

##### Example $$\PageIndex{4}$$

$$\sqrt{4x - 5} = -6$$. By inspection, this equation has no real solution.

The symbol, $$\sqrt{}$$, signifies the positive square root and not the negative square root.

## Practice Set A

Solve each square root equation.

##### Practice Problem $$\PageIndex{1}$$

$$\sqrt{y} = 14$$

$$y = 196$$

##### Practice Problem $$\PageIndex{2}$$

$$\sqrt{a - 7} = 5$$

$$a = 32$$

##### Practice Problem $$\PageIndex{3}$$

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

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

##### Practice Problem $$\PageIndex{4}$$

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

No real solution

## Exercises

For the following problems, solve the square root equations.

##### Exercise $$\PageIndex{1}$$

$$\sqrt{x} = 5$$

$$x = 25$$

##### Exercise $$\PageIndex{2}$$

$$\sqrt{y} = 7$$

##### Exercise $$\PageIndex{3}$$

$$\sqrt{a} = 10$$

$$a = 100$$

##### Exercise $$\PageIndex{4}$$

$$\sqrt{c} = 12$$

##### Exercise $$\PageIndex{5}$$

$$\sqrt{x} = -3$$

No solution

##### Exercise $$\PageIndex{6}$$

$$\sqrt{y} = -6$$

##### Exercise $$\PageIndex{7}$$

$$\sqrt{x} = 0$$

$$x = 0$$

##### Exercise $$\PageIndex{8}$$

$$\sqrt{x} = 1$$

##### Exercise $$\PageIndex{9}$$

$$\sqrt{x + 3} = 3$$

$$x = 6$$

##### Exercise $$\PageIndex{10}$$

$$\sqrt{y - 5} = 5$$

##### Exercise $$\PageIndex{11}$$

$$\sqrt{a + 2} = 6$$

$$a = 34$$

##### Exercise $$\PageIndex{12}$$

$$\sqrt{y + 7} = 9$$

##### Exercise $$\PageIndex{13}$$

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

$$y = 20$$

##### Exercise $$\PageIndex{14}$$

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

##### Exercise $$\PageIndex{15}$$

$$\sqrt{x - 16} = 0$$

$$x = 16$$

##### Exercise $$\PageIndex{16}$$

$$\sqrt{y -25} = 0$$

##### Exercise $$\PageIndex{17}$$

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

$$m = 3$$

##### Exercise $$\PageIndex{18}$$

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

##### Exercise $$\PageIndex{19}$$

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

No solution

##### Exercise $$\PageIndex{20}$$

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

##### Exercise $$\PageIndex{21}$$

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

$$a = 2$$

##### Exercise $$\PageIndex{22}$$

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

##### Exercise $$\PageIndex{23}$$

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

No solution

##### Exercise $$\PageIndex{24}$$

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

##### Exercise $$\PageIndex{25}$$

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

$$m = 4$$

##### Exercise $$\PageIndex{26}$$

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

##### Exercise $$\PageIndex{27}$$

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

No solution

##### Exercise $$\PageIndex{28}$$

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

##### Exercise $$\PageIndex{29}$$

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

No solution

##### Exercise $$\PageIndex{30}$$

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 $$520$$

##### Exercise $$\PageIndex{31}$$

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.

a) $$S = 175$$

b) $$E = 190$$

##### Exercise $$\PageIndex{32}$$

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

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

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 $$\pi = 3.14$$.

##### Exercise $$\PageIndex{33}$$

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

$$F = \dfrac{mm'}{r^2}$$:

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

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

a) $$F = 32$$

b) $$r = 8$$cm.

##### Exercise $$\PageIndex{34}$$

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.

##### Exercise $$\PageIndex{35}$$

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

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 if $$10.8772$$ seconds. Use $$\pi = 3.14$$.

a) $$T = 1.57$$sec

b) $$L = 95.99$$cm

##### Exercise $$\PageIndex{36}$$

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 $$KE = \dfrac{1}{2}mv^2$$.

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

##### Exercise $$\PageIndex{37}$$

Write $$\dfrac{x^{10}y^3(x+7)^4}{x^{-2}y^3(x+7)^{-1}}$$ so that only positive exponents appear.

$$x^{12}(x+7)^5$$

##### Exercise $$\PageIndex{38}$$

Classify $$x+4 = x+7$$ as an identity, a contradiction, or a conditional equation.

##### Exercise $$\PageIndex{39}$$

Supply the missing words. In the coordinate plane, lines with _____ slope rise and lines with _____ slope fall.

positive, negative

##### Exercise $$\PageIndex{40}$$

Simplify $$\sqrt{(x+3)^4(x-2)^6}$$

##### Exercise $$\PageIndex{41}$$

Simplify $$(3 + \sqrt{5})(4 - \sqrt{5})$$

$$7 + \sqrt{5}$$