Radical

Functions

Chapter 13

Solving Radical Equations

Section 13.5

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 3

Examples of rational equations in one variable:

If A and B are expressions, then all solutions of the equation A = B are among the solutions of the equation A2 = B2. That is, the solutions of an equation are among the solutions of the equation obtained by squaring both sides.

Solving Radical Equations in One VariableSquaring Property of Equality

Examples

Property

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 4

Extraneous Solutions:

Because squaring both sides of a square root equation may introduce extraneous solutions, it is essential to check that each proposed solution satisfies the original equation.

Solving Radical Equations in One VariableChecking Proposed Solutions

Property

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 5

Solve the equation

Solving Radical Equations in One VariableSolving a Square Root Equation

Example2 5 13.x

Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 6

Check that 16 satisfies the original equation

Solving Radical Equations in One VariableSolving a Square Root Equation

Solution Continued

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 7

Solve

Solving Radical Equations in One VariableSolving a Square Root Equation

Example1 3.x x

Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 8

Check that 2 and 5 satisfy the original equation

Solving Radical Equations in One VariableSolving a Square Root Equation

Solution Continued

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 9

Solve

Solving Radical Equations in One VariableSolving a Square Root Equation

Example2 3 0.x x

Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 10

Solution Continued

Solving Radical Equations in One VariableSolving a Square Root Equation

Warning

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 11

Solve

Solving Radical Equations in One VariableSolving a Square Root Equation

Example5 1.t t

Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 12

Solution Continued

Solving Radical Equations in One VariableSolving a Square Root Equation

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 13

Solution Continued

Solving Radical Equations in One VariableSolving a Square Root Equation

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 14

To solve a square root equation in one variable,

1. Isolate a square root term on one side of the equation.

2. Square both sides.

3. Repeat steps 1 and 2 until no square root terms remain.

4. Solve the new equation.

5. Check that each proposed solution satisfies the original equation.

Solving Radical Equations in One VariableSolving a Square Root Equation in One Variable

Process

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 15

Solve

Solving Radical Equations in One VariableSolving a Square Root Equation

Example2 1 2 1.x x

Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 16

Solve

Solving Radical Equations in One VariableSolving a Square Root Equation

Example

Solution

Solution Continued

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 17

Solving Radical Equations in One VariableSolving a Square Root Equation

ExampleSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 18

Solving Radical Equations in One VariableSolving a Square Root Equation

ExampleSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 19

Solve

Use the graph to solve

Solving Radical Equations in One VariableSolving a Square Root Equation

Warning

Example28 3 4.x x x

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 20

• Use the “intersect” feature on a graphing calculator

Solving Radical Equations in One VariableUsing Graphing to Solve a Square Root Equation in One Variable

Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 21

Find the x-intercepts of a graph of

Solving Radical Equations in One VariableFinding x-Intercepts of a Square Root Function

Example

3 4 2.f x x x Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 22

• Check 3 is solution

Solving Radical Equations in One VariableFinding x-Intercepts of a Square Root Function

ExampleSolution Continued

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 23

If A and B are expressions and n is a counting number greater than 1, then all solutions of the equation A = B are among the solutions of the equation An = Bn. That is, the solutions of an equation are among the solutions of the equation obtained by raising both sides to the nth power.

Solve

Solving Radical EquationsPower Property of Equality

Property

Example3 2 4 7 9.w

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 24

Solving Radical EquationsSolving a Radical Model

Solution

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 25

Solving Radical EquationsSolving a Radical Model

SolutionCheck:

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 26

Using Radical Model to Make Predictions About the Independent Variable

Using a Radical Model to Make Predictions

ExampleIn Example 10 of Section 9.1, we worked with the model . , where f (t) is the percentage of colleges and universities that offer distance learning at t years since 1997 (see the table). Predict when all colleges and universities will offer distance learning.

17 44f t t

Lehmann, Elementary and Intermediate Algebra, 1edSection 13.5 Slide 27

Using Radical Model to Make Predictions About the Independent Variable

Using a Radical Model to Make Predictions

Solution