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.


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.


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 $y = \frac{4}{3}$ .


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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9.6 Solve equations with square roots

Solve radical equations

Radical equation

Solution

How to solve radical equations

Solution

Solve a radical equation.

Solution

Solution

Solution

Binomial squares

Solution

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