Chapter 8 Quadratic Functions Copyright 2015 2011 2007

Chapter 8 Quadratic Functions Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -

Chapter Sections 8. 1 – Solving Quadratic Equations by Completing the Square 8. 2 – Solving Quadratic Equations by the Quadratic Formulas 8. 3 – Quadratic Equations: Applications and Problem Solving 8. 4 – Writing Equations in Quadratic Form 8. 5 – Graphing Quadratic Functions 8. 6 – Quadratic and Other Inequalities in One Variable Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -2 2

§ 8. 4 Writing Equations in Quadratic Form Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -

Solve Equations That Are Quadratic in Form Equations Quadratic in Form An equation that can be rewritten in the form au 2 + bu + c = 0 for a ≠ 0, where u is an algebraic expression, is called quadratic in form. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -4 4

Solve Equations That Are Quadratic in Form To Solve Equations Quadratic in Form 1. Make a substitution that will result in an equation of the form au 2 + bu + c = 0 for a ≠ 0 where u is a function of the original variable. 2. Solve the equation au 2 + bu + c = 0 for u. 3. Replace u with the function of the original variable from step 1. 4. Solve the resulting equation for the original variable. 5. If, during step 4, you raise both sides of the equation to an even power, check for extraneous solutions by substituting the apparent solutions into the original equation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -5 5

Solve Equations That Are Quadratic in Form Example Solve x 4 – 5 x 2 + 4 = 0. To obtain an equation quadratic in form, we will let u = x 2. Then u 2 = (x 2)2 = x 4. We now have a quadratic equation we can solve by factoring continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -6 6

Solve Equations That Are Quadratic in Form Next, we replace u with x 2 and solve for x. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -7 7

Solve Equations with Rational Exponents Whenever you raise both sides of an equation to an even power, you must check all apparent solutions in the original equation. Example Solve . We will let u=x 1/4, then u 2 = (x 1/4)2 = x 2/4 = x 1/2 continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -8 8

Solve Equations with Rational Exponents Now substitute x 1/4 for u and raise both sides of the equation to the fourth power and solve for x. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -9 9

Solve Equations with Rational Exponents The two possible solutions are 81 and 16. However, since we raised both sides of an equation to an even power, we need to check for extraneous solutions. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -10 10

Solve Equations with Rational Exponents Since 81 does not check, it is an extraneous solution. The only solution is 16. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -11 11