9.7: Square Root Equations with Applications

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 $\PageIndex{1}$

$\sqrt{y} = 14$

Answer

$y = 196$

Practice Problem $\PageIndex{2}$

$\sqrt{a - 7} = 5$

Answer

$a = 32$

Practice Problem $\PageIndex{3}$

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

Answer

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

Practice Problem $\PageIndex{4}$

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

Answer

No real solution

Exercises

For the following problems, solve the square root equations.

Exercise $\PageIndex{1}$

$\sqrt{x} = 5$

Answer

$x = 25$

Exercise $\PageIndex{2}$

$\sqrt{y} = 7$

Exercise $\PageIndex{3}$

$\sqrt{a} = 10$

Answer

$a = 100$

Exercise $\PageIndex{4}$

$\sqrt{c} = 12$

Exercise $\PageIndex{5}$

$\sqrt{x} = -3$

Answer

No solution

Exercise $\PageIndex{6}$

$\sqrt{y} = -6$

Exercise $\PageIndex{7}$

$\sqrt{x} = 0$

Answer

$x = 0$

Exercise $\PageIndex{8}$

$\sqrt{x} = 1$

Exercise $\PageIndex{9}$

$\sqrt{x + 3} = 3$

Answer

$x = 6$

Exercise $\PageIndex{10}$

$\sqrt{y - 5} = 5$

Exercise $\PageIndex{11}$

$\sqrt{a + 2} = 6$

Answer

$a = 34$

Exercise $\PageIndex{12}$

$\sqrt{y + 7} = 9$

Exercise $\PageIndex{13}$

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

Answer

$y = 20$

Exercise $\PageIndex{14}$

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

Exercise $\PageIndex{15}$

$\sqrt{x - 16} = 0$

Answer

$x = 16$

Exercise $\PageIndex{16}$

$\sqrt{y -25} = 0$

Exercise $\PageIndex{17}$

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

Answer

$m = 3$

Exercise $\PageIndex{18}$

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

Exercise $\PageIndex{19}$

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

Answer

No solution

Exercise $\PageIndex{20}$

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

Exercise $\PageIndex{21}$

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

Answer

$a = 2$

Exercise $\PageIndex{22}$

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

Exercise $\PageIndex{23}$

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

Answer

No solution

Exercise $\PageIndex{24}$

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

Exercise $\PageIndex{25}$

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

Answer

$m = 4$

Exercise $\PageIndex{26}$

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

Exercise $\PageIndex{27}$

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

Answer

No solution

Exercise $\PageIndex{28}$

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

Exercise $\PageIndex{29}$

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

Answer

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.

Answer

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 $0.0001$.

Answer

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 $\pi = 3.14$.

Answer

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.

Answer

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

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.

Answer

positive, negative

Exercise $\PageIndex{40}$

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

Exercise $\PageIndex{41}$

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

Answer

$7 + \sqrt{5}$