§ 7.6 Radical Equations.

Slides:



Advertisements
Similar presentations
§ 7.7 Complex Numbers.
Advertisements

§ 7.2 Rational Exponents.
§ 1.4 Solving Linear Equations.
§ 9.5 Exponential and Logarithmic Equations.
HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2010 Hawkes Learning Systems. All rights reserved. Hawkes Learning Systems: College Algebra.
Chapter 15 Roots and Radicals.
College Algebra: Class 4 Radical Equations Objectives: Solve radical equations Solve equations quadratic in form Solve equations by factoring.
Solving Radical Equations and Inequalities 5-8
Remember! For a square root, the index of the radical is 2.
§ 10.5 Systems of Nonlinear Equations in Two Variables.
10.6 Solving Equations with Radicals
Multiplying, Dividing, and Simplifying Radicals
Roots and Radicals.
Section 6.6: Solve Radical Equations Starter: CC 6.5 Part 2.
Unit 1 Expressions, Equations and Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc Other Types of Equations.
TH EDITION Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 1 Equations and Inequalities Copyright © 2013, 2009, 2005 Pearson Education,
Chapter 1 Equations, Inequalities, and Mathematical Models
§ 7.3 Multiplying and Simplifying Radical Expressions.
Solving Radical Equations Module 14 Topic 4.
Entrance Slip: Quadratics 1)2) 3)4) 5)6) Complete the Square 7)
§ 7.6 Radical Equations. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.6 Radical Equations A radical equation is an equation in which the variable.
Chapter 8 Roots and Radicals.
5.7 Polynomial Equations and Their Applications.
6.5 Solving Square Root and Other Radical Equations p390.
1.4 Solving Linear Equations. Blitzer, Algebra for College Students, 6e – Slide #2 Section 1.4 Linear Equations Definition of a Linear Equation A linear.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 8 Rational Exponents, Radicals, and Complex Numbers.
Chapter 8 Section 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
§ 7.6 Radical Equations. Blitzer, Intermediate Algebra, 4e – Slide #79 Solving Radical Equations Solving Radical Equations Containing nth Roots 1) If.
Other Types of Equations. Solving a Polynomial Equation by Factoring 1.Move all terms to one side and obtain zero on the other side. 2.Factor. 3. Apply.
Section 1.6 Other Types of Equations. Polynomial Equations.
Dear Power point User, This power point will be best viewed as a slideshow. At the top of the page click on slideshow, then click from the beginning.
Tom Worthing Intermediate Algebra All rights reserved. 1 Higher order equations Use Factoring method or calculator Know how many roots you will have Determine.
Start Up Day 38 12/15 Solve Each Equation:. OBJECTIVE: ESSENTIAL QUESTION: What is a Radical Equation? When you square each side of an equation, is the.
Chapter P Prerequisites: Fundamental Concepts of Algebra 1 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 P.7 Equations.
Section 6Chapter 8. 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objectives Solving Equations with Radicals Solve radical equations by.
Warm Up Simplify each expression. Assume all variables are positive
Section P7 Equations. Solving Rational Equations.
Radical Equations and Problem Solving Use the power rule to solve radical equations.
Algebra 2 Solving Radical Equations Section 7-5 Solving Square Root and Other Radical Equations Lesson 7-5.
Holt McDougal Algebra 2 Solving Radical Equations and Inequalities Solving Radical Equations and Inequalities Holt Algebra 2 Skills Check Skills Check.
§ 2.2 The Multiplication Property of Equality. Blitzer, Introductory Algebra, 5e – Slide #2 Section 2.2 Properties of Equality PropertyDefinition Addition.
Topic VIII: Radical Functions and Equations 8.1 Solving Radical Equations.
Chapter 8 Section 6. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Equations with Radicals Solve radical equations having.
Section 2.6 – Other Types of Equations
Aim #1.6: How do we solve other types of equations?
Solve Radical Equations
Objective Solve radical equations..
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Solving Radical Equations
Section 1.6 Other Types of Equations
Solving Radical Equations and Inequalities 5-8
Solving Equations Containing
4A.3 - Solving Radical Equations and Inequalities
3-8 Solving Radical equations
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Solving Radical Equations
§ 7.6 Radical Equations.
Solving Equations Containing
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Roots, Radicals, and Complex Numbers
Precalculus Essentials
Solving Equations Containing
Copyright © 2017, 2013, 2009 Pearson Education, Inc.
1.4 Solving Equations Using a Graphing Utility
Chapter 15 Roots and Radicals.
Objective Solve radical equations.. Objective Solve radical equations.
Dear Power point User, This power point will be best viewed as a slideshow. At the top of the page click on slideshow, then click from the beginning.
Chapter 8 Section 6.
Solving Equations Containing
Section 1.6 Other Types of Equations
Presentation transcript:

§ 7.6 Radical Equations

Radical Equations A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root. In this section, you will learn how to solve radical equations. When the variable occurs in a square root, it is necessary to square both sides of the equation. When you square both sides of an equation, sometimes extra answers creep in, called extraneous roots. For example, consider the following very simple original equation. Square both sides. Solve the equation. But -2 does not work in the original equation. Then -2 is an extraneous root. It’s like a hitchhiker that we picked up when we squared both sides. It works only in the squared form and is not a root of the original. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.6

Radical Equations Just to note then…. When you solve a rational equation and must raise both sides to an even power, remember to check your roots. Throw out any extra (extraneous) roots from your solution set. Another thing that you should know is … if your equation contains two or more square root expressions, you will need to isolate one square root expression, square both sides, and then you may have to repeat the process. You may have to square both sides twice. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.6

Solving Radical Equations Containing nth Roots 1) If necessary, arrange terms so that one radical is isolated on one side of the equation. 2) Raise both sides of the equation to the nth power to eliminate the nth root. 3) Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2. 4) Check all proposed solutions in the original equation. Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.6

Solving Radical Equations EXAMPLE Solve: SOLUTION 1) Isolate a radical on one side. The radical, , can be isolated by subtracting 11 from both sides. We obtain 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.6

Solving Radical Equations CONTINUED 3) Solve the resulting equation. This is the equation from step 2. Subtract 5 from both sides. Divide both sides by 2. 4) Check the proposed solution in the original equation. Check 10: ? ? false Therefore there is no solution to the equation. ? Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.6

Solving Radical Equations EXAMPLE Solve: SOLUTION 1) Isolate a radical on one side. The radical, , can be isolated by subtracting 3x from both sides. We obtain 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides. Simplify. Use the special formula Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.6

Solving Radical Equations CONTINUED 3) Solve the resulting equation. Because of the -term, the resulting equation is a quadratic equation. We need to write this quadratic equation in standard form. We can obtain zero on the left side by subtracting 3x and 7 from both sides. Equation from step 2. Subtract 3x and 7 from both sides. Factor out the GCF, 9. Divide both sides by 9. Factor the right side. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.6

Solving Radical Equations CONTINUED Set each factor equal to 0. Solve for x. 4) Check the proposed solutions in the original equation. Check -2: Check -1: ? ? ? ? ? ? false true The solution is -1. The solution set is {-1}. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.6

Solving Radical Equations EXAMPLE Solve: SOLUTION 1) Isolate a radical on one side. The radical, , can be isolated by subtracting from both sides. We obtain 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides. Simplify. Use the special formula Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.6

Solving Radical Equations CONTINUED Combine like terms. 1) Isolate a radical on one side. The radical, can be isolated by subtracting 20 + x from both sides and then dividing both sides by -8. We obtain 2) Raise both sides to the nth power. Because n, the index, is 2, we square both sides. Square the 3 and the Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.6

Solving Radical Equations CONTINUED 3) Solve the resulting equation. This is the equation from the last step. Subtract 4 from both sides. 3) Check the proposed solution in the original equation. Check 5: true ? The solution is 5. The solution set is {5}. ? ? Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.6

Solving Radical Equations EXAMPLE Solve: SOLUTION Although we can rewrite the equation in radical form it is not necessary to do so. Because, the equation involves a fourth root, we isolate the radical term – that is, the term with the rational exponent – and raise both sides to the 4th power. This is the given equation. Subtract 7 from both sides. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.6

Solving Radical Equations CONTINUED Raise both sides to the 4th power. Multiply exponents on the left sides and then simplify. Subtract 3 from both sides. Divide both sides by 2. Upon checking the proposed solution, 39, in the original equation, we find that it checks out and is a solution. Therefore the solution set is {39}. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.6

Solving Radical Equations EXAMPLE For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The function models the number of Earth days in a planet’s year, f (x), where x is the average distance of the planet from the sun, in millions of kilometers. Use the function to solve the following problem. There are approximately 88 Earth days in the year of the planet Mercury. What is the average distance of Mercury from the sun? Use a calculator and round to the nearest million kilometers. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.6

Solving Radical Equations CONTINUED SOLUTION To find the average distance of Mercury from the sun, replace f (x) in the function with 88. This is the given equation. Replace f (x) with 88. Divide both sides by 0.2. Square both sides. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.6

Solving Radical Equations CONTINUED Take the cube root of both sides. Simplify. The model indicates that the average distance, to the nearest million kilometers, that Mercury is from the sun is 58 million kilometers. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.6

In Summary… Solving Radical Equations Containing nth Roots Isolate one radical on one side of the equation. Raise both sides to the nth power Solve the resulting equation Check proposed solutions in the original equation. Sometimes proposed solutions will work in the final simplified form of the original equation, but will not work in the original equation itself. These imposter roots that sometimes slip in when we square both sides of an equation are called extraneous roots. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.6