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. 1 Solving Quadratic Equations by Completing the Square Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -

Quadratic Equations A quadratic equation is an equation of the form ax 2 + bx + c = 0 where a, b, and c are real numbers and a 0. In Section 5. 8 we solved quadratic equations by factoring. In this section we introduce two additional procedures used to solve quadratic equations: the square root property and completing the square. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -4 4

Square Root Property If x 2 = a, where a is a real number, then x = ± √a. Solve the equation x 2 = 49. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -5 5

Square Root Property Solve the equation x 2 - 9 = 0. Solve the equation x 2 + 10 = 85. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -6 6

Understand Perfect Square Trinomials A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. x 2 – 10 x + 25 = (x – 5)2 a 2 + 8 a + 16 = (a + 4)2 p 2 – 14 p + 49 = (p – 7)2 Note that in every perfect square trinomial, the constant term is the square of one-half the coefficient of the x-term. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -7 7

Completing the Square To Solve a Quadratic Equation by Completing the Square 1. Use the multiplication (or division) property of equality, if necessary, to make the leading coefficient 1. 2. Rewrite the equation with the constant by itself on the right side of the equation. 3. Take one-half the numerical coefficient of the first-degree term, square it, and add this quantity to both sides of the equation. 4. Factor the trinomial as the square of a binomial. 5. Use the square root property to take the square root of both sides of the equation. 6. Solve for the variable. 7. Check your solutions in the original equation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -8 8

Completing the Square Example Solve the equation x 2 + 6 x + 5 = 0 by completing the square. Step 1 Since the leading coefficient is 1, step 1 is not necessary. Step 2 Subtract 5 from both sides of the equation. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -9 9

Completing the Square Step 3 Determine the square of one-half the numerical coefficient of the first degree term, 6. Add this value to both sides of the equation. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -10 10

Completing the Square Step 4 By following this procedure, we produce a perfect square trinomial on the left side of the equation. The expression x 2 + 6 x + 9 is a perfect square trinomial that can be factored as (x + 3)2. Step 5 Use the square root property. continued Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -11 11

Completing the Square Step 6 Finally, solve for x by subtracting 3 from both sides of the equation. Step 7 Check both solutions in the original equation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8 -12 12