Radical Equations and Extraneous Roots

Alberta Grade 11 Mathematics

Number and Logic

Isolate the radical, solve, and check every proposed root

A radical is a symbol that asks for a root. For example,

\sqrt{25}=5

because

5^2=25

, and

\sqrt[3]{8}=2

because

2^3=8

. In this lesson, you will solve equations containing square roots or cube roots. The key is to undo the root with the matching power, then check the answer in the original equation. Checking matters because squaring can produce a value that does not satisfy the starting equation.

What you will learn

Recognize square-root and cube-root equations.
Solve equations by isolating the radical and using the matching power.
Check proposed roots in the original equation.
Explain why checking is important, especially after squaring.

1. Review: roots and restrictions

A square root asks for the nonnegative number that squares to the value inside the radical. Thus

\sqrt{36}=6

, not

-6

. A cube root asks for the number that cubes to the value inside. Thus

\sqrt[3]{-8}=-2

. Unlike a square root, a cube root can have a negative value inside.

The expression inside a radical is called the radicand. In a real-number square-root equation, the radicand must be zero or positive. For example,

\sqrt{x-4}

requires

x-4\geq 0

, so

x\geq 4

. A cube-root radicand has no such restriction.

You already know how to undo a square by taking a square root, and how to undo a cube by taking a cube root. When solving an equation, use inverse operations on both sides to keep the equation balanced. For a radical equation, first try to leave the radical by itself on one side.

\sqrt{u}

=v\Rightarrow u=v^2 and v≥ 0

A real square root has a nonnegative value and a nonnegative radicand.
A real cube root can have a negative or positive radicand.
Check the original equation, not just a rearranged equation.

2. Solving square-root equations

Suppose a square root is isolated, as in

\sqrt{A}=B

. Squaring both sides removes the square-root symbol and gives

A=B^2

. The condition

B\geq 0

is important: a square root cannot equal a negative number.

After solving the resulting equation, substitute each proposed value into the original equation. Squaring can change an equation into one that has extra solutions. A value that works after squaring may fail in the original equation.

If a square root is not isolated, use inverse operations first. For example, in

\sqrt{x+2}+3=8

, subtract

3

from both sides before squaring. This keeps the steps clear and makes it easier to identify the restriction.

\sqrt{A}=B\Rightarrow A=B^2

Isolate the square root before squaring.
Keep track of the requirement that the isolated right side is nonnegative.
Substitution into the original equation confirms whether a proposed value is a solution.

3. Solving cube-root equations

A cube root is undone by cubing both sides. If

\sqrt[3]{A}=B

, then

A=B^3

. Cubing both sides keeps negative and positive values possible, so there is no extra nonnegative condition on

B

.

As with square-root equations, isolate the radical first. Then solve the resulting equation using familiar operations. Finally, substitute the proposed value into the original equation. This check confirms that arithmetic and signs are correct.

A quick estimate can also help. If a cube root is equal to

3

, its radicand must be

27

, because

3^3=27

. If the radicand in the equation does not become

27

after substitution, the proposed value is not correct.

\sqrt[3]{A}=B\Rightarrow A=B^3

Isolate the cube root and cube both sides.
Cube roots are defined for negative radicands as well as positive ones.
Check the value in the equation as it was originally written.

4. A dependable solving routine

Use a short routine for either kind of equation. First, identify whether the radical is a square root or cube root. Second, isolate it using inverse operations. Third, apply the matching power. Fourth, solve the equation that remains. Last, check every proposed value in the original equation.

For a square-root equation, also make sure the expression inside the square root is nonnegative and that the isolated right side is not negative. These conditions can rule out a value before substitution, but they do not replace the final check.

A solution is a value that makes the original equation true. A proposed root is a value you are testing. Do not call it a solution until the check works.

Match the operation to the radical: square for square root, cube for cube root.
A solution must satisfy the original equation and its restrictions.
Keep enough working to show how each proposed value was obtained.

Worked example

Square-root equation with an extra proposed value

Solve

\sqrt{x+5}=x-1

and check the proposed values.

Check the square-root condition
The radicand must be nonnegative, so $x+5\geq 0$ . Also, the right side must be nonnegative because it equals a square root. Thus $x-1\geq 0$ , giving $x\geq 1$ .
$x\geq 1$
Square both sides
The radical is isolated. Squaring both sides removes it, but the resulting equation may include a value that fails the original equation.
$x+5=(x-1)^2$
Solve the resulting equation
Expand the square, collect terms on one side, and factor. The two proposed values are $4$ and $-1$ .
$x^2-3x-4=0\Rightarrow (x-4)(x+1)=0$
Check both proposed values
For $x=4$ , the original equation becomes $3=3$ , so $4$ works. For $x=-1$ , the radicand is $4$ , but the right side is $-2$ ; the equation would say $2=-2$ . Also, $-1$ does not meet $x\geq 1$ . $\sqrt{4+5}$ =4-1 and $\sqrt{-1+5}$ \ne -1-1

Answer: The solution is

x=4

.

Check: Substitution in the original equation gives

\sqrt{9}=3

, which is true.

Worked example

Cube-root equation with a negative value

Solve

\sqrt[3]{2x-1}=-3

and check the answer.

Cube both sides
The cube root is already isolated. Cubing both sides removes it. A negative answer is allowed because a negative number cubed is negative.
$2x-1=(-3)^3$
Solve for the variable
Add $1$ to both sides, then divide by $2$ to leave $x$ by itself.
$2x-1=-27\Rightarrow 2x=-26\Rightarrow x=-13$
Check in the original equation
Substitute $-13$ for $x$ . The radicand becomes $-27$ , whose cube root is $-3$ . The original equation is true.
$\sqrt[3]{2(-13)-1}=\sqrt[3]{-27}=-3$

Answer: The solution is

x=-13

.

Check: The negative solution is reasonable because cube roots can have negative values.

Worked example

Square-root equation with a number added

Solve

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

and check the answer.

Isolate the square root
Subtract $2$ from both sides. The square root is now by itself and equals a nonnegative number.
$\sqrt{3x+1}=4$
Square and solve
Square both sides, then subtract $1$ and divide by $3$ . The radicand at the resulting value is positive, so it meets the square-root restriction.
$3x+1=16\Rightarrow 3x=15\Rightarrow x=5$
Check the original equation
Substitute $5$ into the original expression. The left side equals $6$ , so the proposed value is a solution.
$\sqrt{3(5)+1}+2=\sqrt{16}+2=6$

Answer: The solution is

x=5

.

Check: The substitution returns the original right side,

6

.

Common mistakes and how to avoid them

Squaring before isolating the square root.

Correction: Use inverse operations to get the radical alone first. This reduces errors and makes the check easier.

Accepting every value found after squaring.

Correction: Squaring can create an extra proposed value. Substitute each value into the original equation.

Rejecting a negative cube-root value.

Correction: A cube root can be negative. Check the value by cubing it or by substitution.

Checking only the radicand and not the full equation.

Correction: A nonnegative radicand is required for a square root, but the complete original equation must also be true.

Lesson summary

Isolate the radical before using a power.
Square both sides to undo a square root, and cube both sides to undo a cube root.
For a real square root, the radicand and the isolated right side must be nonnegative.
Check every proposed value in the original equation before calling it a solution.

Check your understanding

Question 1

Solve

\sqrt{x+4}=5

. Which value is the solution?

$x=21$
$x=29$
$x=-29$
$x=1$

Show answer and explanation

x=29

Squaring gives

x+4=25

, so

x=21

—correction: the matching option is

x=21

. Substitution gives

\sqrt{25}=5

.

Question 2

Solve

\sqrt[3]{x-6}=2

. Which value is the solution?

$x=2$
$x=8$
$x=14$
$x=12$

Show answer and explanation

x=14

Cube both sides to get

x-6=8

, so

x=14

. Checking gives

\sqrt[3]{14-6}=\sqrt[3]{8}=2

.

Key terms

Radical: A symbol that indicates a root, such as a square root or cube root.
Radicand: The number or expression inside a radical.
Proposed root: A value being tested to see whether it solves an equation.
Solution: A value that makes the original equation true.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 11 Mathematics (Math 20-2), expectation NL4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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NL4 · Radical Equations and Extraneous Roots

Alberta Grade 11 Mathematics

Isolate the radical, solve, and check every proposed root

What you will learn

1. Review: roots and restrictions

2. Solving square-root equations

3. Solving cube-root equations

4. A dependable solving routine

Square-root equation with an extra proposed value

Cube-root equation with a negative value

Square-root equation with a number added

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Math 20-2

Tutor support for NL4 · Solve square-root and cube-root equations and check proposed roots

About this lesson and its review