1 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-1 Quadratic Functions Chapter 8

2 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter – 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 Chapter Sections

3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-3 § 8.2 Solving Quadratic Equations by the Quadratic Formula

4 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-4 The quadratic formula can be used to solve any quadratic equation. It is the most versatile method of solving quadratic equations.. Quadratic Formula To use the quadratic formula, the equation must be written in standard form ax 2 + bx + c = 0, where a is the coefficient of the squared term, b is the coefficient of the first-degree term, and c is the constant. 3x 2 + 4x – 5 = 0 a = 3, b = 4, and c = – 5

5 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-5 Quadratic Formula To Solve a Quadratic Equation by the Quadratic Formula 1.Write the quadratic equation in standard form, ax 2 + bx + c = 0, and determine the numerical values for a, b, and c. 2.Substitute the values for a, b, and c into the quadratic formula and then evaluate the formula to obtain the solution.

6 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-6 Quadratic Formula Example Solve x 2 + 2x – 8 = 0 by using the quadratic formula. continued

7 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-7 Quadratic Formula or

8 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-8 Example Determine an equation that has the solutions -5 and 1. Determine a Quadratic Equation Given Its Solutions

9 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-9 Discriminant The discriminant of a quadratic equation is the expression under the radical sign in the quadratic formula.

10 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-10 Solutions of a Quadratic Equation For a quadratic equation of the form ax 2 + bx + c = 0, a ≠ 0: If b 2 – 4ac > 0, the quadratic equation has two distinct real number solutions. If b 2 – 4ac = 0, the quadratic equation has a single real number solution. If b 2 – 4ac < 0, the quadratic equation has no real number solution.

11 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter If b 2 – 4ac > 0, f(x) has two distinct x-intercepts. Graphs of f(x) = ax 2 + bx + c y x or y x Discriminant

12 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter If b 2 – 4ac = 0, f(x) has one single x-intercept. Graphs of f(x) = ax 2 + bx + c y x or y x Discriminant

13 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter If b 2 – 4ac < 0, f(x) has no x-intercept. Graphs of f(x) = ax 2 + bx + c y x or y x Discriminant

14 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-14 Study Applications That Use Quadratic Equations Example Mary Olson owns a business that sells cell phones. The revenue, R(n), from selling the cells phones is determined by multiplying the number of cell phones by the selling price per phone. Suppose the revenue from selling n cell phone, n ≤ 50, is R(n)=n(50-0.2n) where (50-0.2n) is the price per cell phone, in dollars. Find the revenue when 30 cell phones are sold. continued

15 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 8-15 Study Applications That Use Quadratic Equations To find the revenue when 30 cell phones are sold, we evaluate the revenue function for n = 30. The revenue from selling 30 cell phones is $1,320.