Want to create or adapt OER like this? Learn how BCcampus supports open education and how you can access Pressbooks . Learn more about how Pressbooks supports open publishing practices. -->

Roots and Radicals

Solve Radical Equations

## Learning Objectives

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

Before you get started, take this readiness quiz.

- Isolate the radical on one side of the equation.
- Raise both sides of the equation to the power of the index.
- Solve the new equation.
- Check the answer in the original equation.

Because the square root is equal to a negative number, the equation has no solution.

When the index of the radical is 3, we cube both sides to remove the radical.

Solve Radical Equations with Two Radicals

In the next example, when one radical is isolated, the second radical is also isolated.

If yes, repeat Step 1 and Step 2 again.

- Read the problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.
- Identify what we are looking for.
- Name what we are looking for by choosing a variable to represent it.
- Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.

It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.

- Solving an Equation Involving a Single Radical
- Solving Equations with Radicals and Rational Exponents
- Solving Radical Equations
- Radical Equation Application

## Key Concepts

## Practice Makes Perfect

In the following exercises, solve.

In the following exercises, solve. Round approximations to one decimal place.

## Writing Exercises

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

## Share This Book

## Solving Radical Equations

How to solve equations with square roots, cube roots, etc.

## Radical Equations

We can get rid of a square root by squaring (or cube roots by cubing, etc).

Then continue with our solution!

## Example: solve √(2x+9) − 5 = 0

Now it should be easier to solve!

Check: √(2·8+9) − 5 = √(25) − 5 = 5 − 5 = 0

## More Than One Square Root

What if there are two or more square roots? Easy! Just repeat the process for each one.

It will take longer (lots more steps) ... but nothing too hard.

## Example: solve √(2x−5) − √(x−1) = 1

We have removed one square root.

Now do the "square root" thing again:

We have now successfully removed both square roots.

Let us continue on with the solution.

It is a Quadratic Equation! So let us put it in standard form.

Using the Quadratic Formula (a=1, b=−14, c=29) gives the solutions:

2.53 and 11.47 (to 2 decimal places)

There is really only one solution :

Answer: 11.47 (to 2 decimal places)

See? This method can sometimes produce solutions that don't really work!

The root that seemed to work, but wasn't right when we checked it, is called an "Extraneous Root"

- My Preferences
- My Reading List
- Study Guides
- Solving Radical Equations
- Linear Equations
- Quiz: Linear Equations
- Quiz: Formulas
- Absolute Value Equations
- Quiz: Absolute Value Equations
- Linear Inequalities
- Quiz: Linear Inequalities
- Compound Inequalities
- Quiz: Compound Inequalities
- Absolute Value Inequalities
- Quiz: Absolute Value Inequalities
- Rectangular Coordinate System
- Quiz: Rectangular Coordinate System
- Distance Formula
- Quiz: Distance Formula
- Midpoint Formula
- Quiz: Midpoint Formula
- Slope of a Line
- Quiz: Slope of a Line
- Slopes of Parallel and Perpendicular Lines
- Quiz: Slopes of Parallel and Perpendicular Lines
- Equations of Lines
- Quiz: Equations of Lines
- Graphs of Linear Inequalities
- Quiz: Graphs of Linear Inequalities
- Linear Equations: Solutions Using Graphing with Two Variables
- Quiz: Linear Equations: Solutions Using Graphing with Two Variables
- Linear Equations: Solutions Using Substitution with Two Variables
- Quiz: Linear Equations: Solutions Using Substitution with Two Variables
- Linear Equations: Solutions Using Elimination with Two Variables
- Quiz: Linear Equations: Solutions Using Elimination with Two Variables
- Linear Equations: Solutions Using Matrices with Two Variables
- Quiz: Linear Equations: Solutions Using Matrices with Two Variables
- Linear Equations: Solutions Using Determinants with Two Variables
- Quiz: Linear Equations: Solutions Using Determinants with Two Variables
- Linear Inequalities: Solutions Using Graphing with Two Variables
- Quiz: Linear Inequalities: Solutions Using Graphing with Two Variables
- Quiz: Linear Equations: Solutions Using Elimination with Three Variables
- Linear Equations: Solutions Using Elimination with Three Variables
- Linear Equations: Solutions Using Matrices with Three Variables
- Quiz: Linear Equations: Solutions Using Matrices with Three Variables
- Linear Equations: Solutions Using Determinants with Three Variables
- Quiz: Linear Equations: Solutions Using Determinants with Three Variables
- Adding and Subtracting Polynomials
- Quiz: Adding and Subtracting Polynomials
- Multiplying Polynomials
- Quiz: Multiplying Polynomials
- Special Products of Binomials
- Quiz: Special Products of Binomials
- Dividing Polynomials
- Quiz: Dividing Polynomials
- Synthetic Division
- Quiz: Synthetic Division
- Greatest Common Factor
- Quiz: Greatest Common Factor
- Difference of Squares
- Quiz: Difference of Squares
- Sum or Difference of Cubes
- Quiz: Sum or Difference of Cubes
- Trinomials of the Form x^2 + bx + c
- Quiz: Trinomials of the Form x^2 + bx + c
- Trinomials of the Form ax^2 + bx + c
- Quiz: Trinomials of the Form ax^2 + bx + c
- Square Trinomials
- Quiz: Square Trinomials
- Factoring by Regrouping
- Quiz: Factoring by Regrouping
- Summary of Factoring Techniques
- Solving Equations by Factoring
- Quiz: Solving Equations by Factoring
- Examples of Rational Expressions
- Quiz: Examples of Rational Expressions
- Simplifying Rational Expressions
- Quiz: Simplifying Rational Expressions
- Multiplying Rational Expressions
- Quiz: Multiplying Rational Expressions
- Dividing Rational Expressions
- Quiz: Dividing Rational Expressions
- Adding and Subtracting Rational Expressions
- Quiz: Adding and Subtracting Rational Expressions
- Complex Fractions
- Quiz: Complex Fractions
- Solving Rational Equations
- Quiz: Solving Rational Equations
- Proportion, Direct Variation, Inverse Variation, Joint Variation
- Quiz: Proportion, Direct Variation, Inverse Variation, Joint Variation
- Graphing Rational Functions
- Quiz: Graphing Rational Functions
- Basic Definitions
- Quiz: Basic Definitions
- Function Notation
- Quiz: Function Notation
- Compositions of Functions
- Quiz: Compositions of Functions
- Algebra of Functions
- Quiz: Algebra of Functions
- Inverse Functions
- Quiz: Inverse Functions
- Polynomial Function
- Quiz: Polynomial Function
- Remainder Theorem
- Quiz: Remainder Theorem
- Factor Theorem
- Quiz: Factor Theorem
- Zeros of a Function
- Quiz: Zeros of a Function
- Rational Zero Theorem
- Quiz: Rational Zero Theorem
- Graphing Polynomial Functions
- Quiz: Graphing Polynomial Functions
- What Are Radicals?
- Quiz: Radicals
- Simplifying Radicals
- Quiz: Simplifying Radicals
- Adding and Subtracting Radical Expressions
- Quiz: Adding and Subtracting Radical Expressions
- Multiplying Radical Expressions
- Quiz: Multiplying Radical Expressions
- Dividing Radical Expressions
- Quiz: Dividing Radical Expressions
- Rational Exponents
- Quiz: Rational Exponents
- Complex Numbers
- Quiz: Complex Numbers
- Quadratic Equations
- Solving Quadratics by Factoring
- Quiz: Solving Quadratics by Factoring
- Solving Quadratics by the Square Root Property
- Quiz: Solving Quadratics by the Square Root Property
- Solving Quadratics by Completing the Square
- Quiz: Solving Quadratics by Completing the Square
- Solving Quadratics by the Quadratic Formula
- Quiz: Solving Quadratics by the Quadratic Formula
- Solving Equations in Quadratic Form
- Quiz: Solving Equations in Quadratic Form
- Quiz: Solving Radical Equations
- Solving Quadratic Inequalities
- Quiz: Solving Quadratic Inequalities
- The Four Conic Sections
- Quiz: The Four Conic Sections
- Quiz: Circle
- Quiz: Parabola
- Quiz: Ellipse
- Quiz: Hyperbola
- Systems of Equations Solved Algebraically
- Quiz: Systems of Equations Solved Algebraically
- Systems of Equations Solved Graphically
- Quiz: Systems of Equations Solved Graphically
- Systems of Inequalities Solved Graphically
- Exponential Functions
- Quiz: Exponential Functions
- Logarithmic Functions
- Quiz: Logarithmic Functions
- Properties of Logarithms
- Quiz: Properties of Logarithms
- Exponential and Logarithmic Equations
- Quiz: Exponential and Logarithmic Equations
- Definition and Examples of Sequences
- Quiz: Definition and Examples of Sequences
- Arithmetic Sequence
- Quiz: Arithmetic Sequence
- Arithmetic Series
- Quiz: Arithmetic Series
- Geometric Sequence
- Quiz: Geometric Sequence
- Geometric Series
- Quiz: Geometric Series
- Summation Notation
- Quiz: Summation Notation
- Quiz: Factorials
- Binomial Coefficients and the Binomial Theorem
- Quiz: Binomial Coefficients and the Binomial Theorem
- Permutations
- Quiz: Permutations
- Combinations
- Quiz: Combinations
- General Strategy
- Simple Interest
- Quiz: Simple Interest
- Compound Interest
- Quiz: Compound Interest
- Quiz: Mixture
- Quiz: Motion
- Arithmetic/Geometric Series
- Quiz: Arithmetic/Geometric Series
- Algebra II Quiz

Raise both sides of the equation to the index of the radical.

Isolate the radical expression.

Raise both sides to the index of the radical; in this case, square both sides.

This quadratic equation now can be solved either by factoring or by applying the quadratic formula.

Isolate one of the radical expressions.

This is still a radical equation. Isolate the radical expression.

This can be solved either by factoring or by applying the quadratic formula.

Isolate the radical involving the variable.

The check of the solution x = –15 is left to you.

Previous Quiz: Solving Equations in Quadratic Form

Next Quiz: Solving Radical Equations

Are you sure you want to remove #bookConfirmation# and any corresponding bookmarks?

- Practice Problems
- Assignment Problems
- Show all Solutions/Steps/ etc.
- Hide all Solutions/Steps/ etc.
- Equations Reducible to Quadratic in Form
- Linear Inequalities
- Preliminaries
- Graphing and Functions
- Calculus II
- Calculus III
- Differential Equations
- Algebra & Trig Review
- Common Math Errors
- Complex Number Primer
- How To Study Math
- Cheat Sheets & Tables
- MathJax Help and Configuration
- Notes Downloads
- Complete Book
- Practice Problems Downloads
- Complete Book - Problems Only
- Complete Book - Solutions
- Assignment Problems Downloads
- Other Items
- Get URL's for Download Items
- Print Page in Current Form (Default)
- Show all Solutions/Steps and Print Page
- Hide all Solutions/Steps and Print Page

## Section 2.10 : Equations with Radicals

Solve each of the following equations.

- \(2x = \sqrt {x + 3} \) Solution
- \(\sqrt {33 - 2x} = x + 1\) Solution
- \(7 = \sqrt {39 + 3x} - x\) Solution
- \(x = 1 + \sqrt {2x - 2} \) Solution
- \(1 + \sqrt {1 - x} = \sqrt {2x + 4} \) Solution

## Solving Radical Equations

## What is a Radical Equation?

1) Isolate the radical symbol on one side of the equation

2) Square both sides of the equation to eliminate the radical symbol

3) Solve the equation that comes out after the squaring process

4) Check your answers with the original equation to avoid extraneous values

## Examples of How to Solve Radical Equations

Example 1 : Solve the radical equation

Yes, it checks, so x = 16 is a solution.

Example 2 : Solve the radical equation

Looks good for both of our solved values of x after checking, so our solutions are x = 1 and x = 3 .

Example 3 : Solve the radical equation

Example 4 : Solve the radical equation

So for our first step, let’s square both sides and see what happens.

I will leave it to you to check that indeed x = 4 is a solution.

Example 5 : Solve the radical equation

Example 6 : Solve the radical equation

Looking good so far! Now it’s time to square both sides again to finally eliminate the radical.

So the possible solutions are x = 2 , and x = {{ - 22} \over 7} .

Example 7 : Solve the radical equation

The possible solutions then are x = {{ - 5} \over 2} and x = 3 .

You might also be interested in:

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

## Unit 10: Lesson 2

- Intro to square-root equations & extraneous solutions
- Square-root equations intro
- Intro to solving square-root equations
- Solving square-root equations

## Solving square-root equations: one solution

## Want to join the conversation?

## Video transcript

· Solve equations containing radicals.

· Recognize extraneous solutions.

· Solve application problems that involve radical equations as part of the solution.

Solving Application Problems with Radical Equations

## IMAGES

## VIDEO

## COMMENTS

The six steps of problem solving involve problem definition, problem analysis, developing possible solutions, selecting a solution, implementing the solution and evaluating the outcome. Problem solving models are used to address issues that...

The answer to any math problem depends on upon the question being asked. In most math problems, one needs to determine a missing variable. For instance, if a problem reads as 2+3 = , one needs to figure out what the number after the equals ...

To calculate percentages, convert the percentage to a decimal and multiply it by the number in the problem. For example, to find 40 percent of 50, change it to 0.40 times 50, which gives you the result of 20.

This algebra video tutorial explains how to solve radical equations. It contains plenty of examples and practice problems.

Learn how to solve radical equations in this free math video tutorial by Mario's Math Tutoring. We go through 2 different examples.0:19 How

Solve Radical Equations · Isolate the radical on one side of the equation. · Raise both sides of the equation to the power of the index. · Solve the new equation.

A Radical Equation is an equation with a square root or cube root, etc. Solving Radical Equations. We can get rid of a square root by squaring (or cube roots by

Solving Radical Equations · Isolate the radical expression involving the variable. · Raise both sides of the equation to the index of the radical. · If there is

Section 2.10 : Equations with Radicals · 2x=√x+3 2 x = x + 3 Solution · √33−2x=x+1 33 − 2 x = x + 1 Solution · 7=√39+3x−x 7 = 39 + 3 x − x

Solving Radical Equations · 1) Isolate the radical symbol on one side of the equation · 2) Square both sides of the equation to eliminate the radical symbol · 3)

An equation in which a variable is in the radicand of a radical expression is called a radical equation. As usual, when solving these equations

What do I need to know to solve radical equations? · Isolate the radical expression to one side of the equation. · Square both sides the equation. · Rearrange and

We're asked to solve the equation, 3 plus the principal square root of 5x plus 6 is equal to 12. And so the general strategy to solve this type of equation is

A common method for solving radical equations is to raise both sides of an equation to whatever power will eliminate the radical sign from the equation. But be