9.6 Solve equations with square roots

Elementary algebra Page 1 / 6

By the end of this section, you will be able to:

Solve radical equations
Use square roots in applications

Before you get started, take this readiness quiz.

Simplify: ⓐ $\sqrt{9}$ ⓑ $9^{2}$ .
If you missed this problem, review [link] and [link] .
Solve: $5 (x + 1) - 4 = 3 (2 x - 7)$ .
If you missed this problem, review [link] .
Solve: $n^{2} - 6 n + 8 = 0$ .
If you missed this problem, review [link] .

Solve radical equations

In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.

Radical equation

An equation in which the variable is in the radicand of a square root is called a radical equation .

As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are ‘opposite’ operations, we will square both sides in order to remove the radical sign and solve for the variable inside.

But remember that when we write $\sqrt{a}$ we mean the principal square root. So $\sqrt{a} \geq 0$ always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make $\sqrt{a}$ negative. This algebraic solution would not be a solution to the original radical equation ; it is an extraneous solution. We saw extraneous solutions when we solved rational equations, too.

For the equation $\sqrt{x + 2} = x$ :

ⓐ Is $x = 2$ a solution? ⓑ Is $x = −1$ a solution?

Solution

ⓐ Is $x = 2$ a solution?


Let x = 2.
Simplify.

	2 is a solution.

ⓑ Is

x = −1

a solution?


Let x = −1.
Simplify.

	−1 is not a solution.
	−1 is an extraneous solution to the equation.

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For the equation $\sqrt{x + 6} = x$ :

ⓐ Is $x = −2$ a solution? ⓑ Is $x = 3$ a solution?

ⓐ no ⓑ $yes$

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For the equation $\sqrt{− x + 2} = x$ :

ⓐ Is $x = −2$ a solution? ⓑ Is $x = 1$ a solution?

ⓐ no ⓑ $yes$

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Now we will see how to solve a radical equation. Our strategy is based on the relation between taking a square root and squaring.

For a \geq 0, {(\sqrt{a})}^{2} = a

How to solve radical equations

Solve: $\sqrt{2 x - 1} = 7$ .

Solution

This table has three columns and four rows. The first row says, “Step 1. Isolate the radical on one side of equation. The square root of (2x minus 1) is already isolated on the left side.” It then shows the equation: the square root of (2x minus 1) equals 7.

The second row says, “Step 2. Square both sides of the equation. Remember, the square root of a squared equals a.” It then shows the equation: the square root of (2x minus 1) squared equals 7 squared.

The third row then says, “Step 3. Solve the new equation.” It indicates that 2x minus 1 equals 49 or 2x equals 50 which means that x equals 25.

The fourth row says, “Step 4. Check the answer. Check:” It then indicates the square root of (2x minus 1) equals 7. This becomes the square root of (2 times 25 minus 1) equals 7. This becomes the square root of (50 minus 1) equals 7. This becomes the square root of 49 equals 7, and thus 7 equals 7. The figure then states, “The solutions is x equals 25.”

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Solve: $\sqrt{3 x - 5} = 5$ .

$10$

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Solve: $\sqrt{4 x + 8} = 6$ .

$7$

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Solve a radical equation.

Isolate the radical on one side of the equation.
Square both sides of the equation.
Solve the new equation.
Check the answer.

Solve: $\sqrt{5 n - 4} - 9 = 0$ .

Solution


To isolate the radical, add 9 to both sides.
Simplify.
Square both sides of the equation.
Solve the new equation.


Check the answer.

	The solution is n = 17.

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Solve: $\sqrt{3 m + 2} - 5 = 0$ .

$\frac{23}{3}$

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Solve: $\sqrt{10 z + 1} - 2 = 0$ .

$\frac{3}{10}$

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Solve: $\sqrt{3 y + 5} + 2 = 5$ .

Solution


To isolate the radical, subtract 2 from both sides.
Simplify.
Square both sides of the equation.
Solve the new equation.


Check the answer.

	The solution is $y = \frac{4}{3}$ .

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Solve: $\sqrt{3 p + 3} + 3 = 5$ .

$\frac{1}{2}$

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Solve: $\sqrt{5 q + 1} + 4 = 6$ .

$\frac{3}{5}$

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When we use a radical sign, we mean the principal or positive root. If an equation has a square root equal to a negative number, that equation will have no solution.

Solve: $\sqrt{9 k - 2} + 1 = 0$ .

Solution


To isolate the radical, subtract 1 from both sides.
Simplify.
Since the square root is equal to a negative number, the equation has no solution.

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Solve: $\sqrt{2 r - 3} + 5 = 0$ .

$no solution$

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Solve: $\sqrt{7 s - 3} + 2 = 0$ .

$no solution$

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If one side of the equation is a binomial, we use the binomial squares formula when we square it.

Binomial squares

\begin{matrix} {(a + b)}^{2} = a^{2} + 2 a b + b^{2} & {(a - b)}^{2} = a^{2} - 2 a b + b^{2} \end{matrix}

Don’t forget the middle term!

Solve: $\sqrt{p - 1} + 1 = p$ .

Solution


To isolate the radical, subtract 1 from both sides.
Simplify.
Square both sides of the equation.
Simplify, then solve the new equation.
It is a quadratic equation, so get zero on one side.
Factor the right side.
Use the zero product property.
Solve each equation.
Check the answers.

	The solutions are p = 1, p = 2.

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

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chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

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answer

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progressive wave

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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

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Source: OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2

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