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OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Other Types of Equations Learn to solve equations by factoring. Learn to solve fractional equations. Learn to solve equations involving radicals. Learn to solve equations that are quadratic in form. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR SOLVING EQUATIONS BY FACTORING Step 1Make one side zero. Move all nonzero terms in the equation to one side (say the left side), so that the other side (right side) is 0. Step 2Factor the left side. Step 3Use the zero-product property. Set each factor in Step 2 equal to 0, and then solve the resulting equations. Step 4Check your solutions.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Solving an Equation by Factoring Solve by factoring: The solution set is {–3, 0, 3}. Solution Step 1 Step 2 Step 3 Step 4 

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Solving an Equation by Factoring Solve by factoring: The solution set is {2,i,–i}. Solution Step 1 Step 2 Step 3 Step 4  

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solving a Rational Equation Solve: Solution Step 1Find the LCD: 6x(x + 1) Step 2

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Solving a Rational Equation The solution set is {–3,2}. Solution continued Step 4 Step 3 Step 5Both solutions check in the original equation.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving Equations Involving Radicals Solve: Solution Since we raise both sides to power 2.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Solving Equations Involving Radicals Solution continued –3 is an extraneous solution. The solution set is {0, 2}. Check each solution. ?  ?  ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving Equations Involving Radicals Solve: Solution Step 1Isolate the radical on one side. Step 2Square both sides and simplify.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Solving Equations Involving Radicals Solution continued Step 3Set each factor = 0. 0 is an extraneous solution. The solution set is {4}. Step 4Check. ?  ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Solving an Equation Involving Two Radicals Solve: Solution Step 1Isolate one of the radicals. Step 2Square both sides and simplify.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Solving an Equation Involving Two Radicals Solution continued Step 3Repeat the process - isolate the radical, square both sides, simplify and factor.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Solving an Equation Involving Two Radicals Solution continued Step 4Set each factor = 0. The solution set is {1,5}. Step 5Check.  ?  ?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SOLVING EQUATIONS CONTAINING SQUARE ROOTS Step 1Isolate one radical to one side of the equation. Step 2Square both sides of the equation in Step 1 and simplify. Step 3If the equation in Step 2 contains a radical, repeat Steps 1 and 2 to get an equation that is free of radicals. Step 5Check the solutions in the original equation. Step 4Solve the equation obtained in Steps

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley An equation in a variable x is quadratic in form if it can be written as EQUATIONS THAT ARE QUADRATIC IN FORM where u is an expression in the variable x. We solve the equation au 2 + bu + c = 0 for u. Then the solutions of the original equation can be obtained by replacing u by the expression in x that u represents.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving an Equation That Is Quadratic in Form by Substitution Solve: Solution Let u = x 2 – 1, then u 2 = (x 2 – 1) 2

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Solving an Equation That Is Quadratic in Form by Substitution Solution continued Replace u with x 2 – 1, and solve for x. All four solutions check in the original equation. The solution set is {i, –i, 3, –3}.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solving an Equation That Is Quadratic in Form by Substitution Solve: Solution Let then

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solving an Equation That Is Quadratic in Form by Substitution Solution continued Replace u with and solve for x. x = 1 checks in the original equation

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Solving an Equation That Is Quadratic in Form by Substitution Solution continued Both solutions check in the original equation. The solution set is

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Investigating Space Travel Your sister is 5 years older than you are. She decides she has had enough of Earth and needs a vacation. She takes a trip to the Omega-One star system. Her trip to Omega-One and back in a spacecraft traveling at an average speed v took 15 years, according to the clock and calendar on the spacecraft. But on landing back on Earth, she discovers that her voyage took 25 years, according to the time on Earth. This means that, although you were 5 years younger than your

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Investigating Space Travel sister before her vacation, you are 5 years older than her after her vacation! Use the time- dilation equation from the introduction to this section to calculate the speed of the spacecraft.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Investigating Space Travel Substitute t 0 = 15 (moving-frame time) and t = 25 (fixed-frame time) to obtain Solution So the spacecraft was moving at 80% (0.8c) the speed of light.