Copyright © 2006 Pearson Education, Inc Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Quadratic Functions and Equations 8 Quadratic Functions and Equations 8.1 Quadratic Equations 8.2 The Quadratic Formula 8.3 Applications Involving Quadratic Equations 8.4 Studying Solutions of Quadratic Equations 8.5 Equations Reducible to Quadratic 8.6 Quadratic Functions and Their Graphs 8.7 More About Graphing Quadratic Functions 8.8 Problem Solving and Quadratic Functions 8.9 Polynomial and Rational Inequalities Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solving Using the Quadratic Formula Approximating Solutions 8.2 The Quadratic Formula Solving Using the Quadratic Formula Approximating Solutions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solving Using the Quadratic Formula Each time we solve by completing the square, the procedure is the same. In mathematics, when a procedure is repeated many times, a formula can often be developed to speed up our work. If we begin with a quadratic equation in standard form, ax2 + bx + c = 0, and solve by completing the square we arrive at the quadratic formula. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The Quadratic Formula The solutions of ax2 + bx + c = 0, are given by Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Solve 3x2 + 5x = 2 using the quadratic formula. First determine a, b, and c: 3x2 + 5x – 2 = 0; a = 3, b = 5, and c = –2. Substituting Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The solutions are 1/3 and –2. The check is left to the student. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Solve a Quadratic Equation If the equation can easily be written in the form ax2 = p or (x + k)2 = d, use the principle of square roots. If step (1) does not apply, write the equation in the form ax2 + bx + c = 0. Try factoring using the principle of zero products. If factoring seems difficult or impossible, use the quadratic formula. Completing the square can also be used, but is slower. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Recall that a second-degree polynomial in one variable is said to be quadratic. Similarly, a second-degree polynomial function in one variable is said to be a quadratic function. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Solve x2 + 7 = 2x using the quadratic formula. First determine a, b, and c: x2 – 2x + 7 = 0; a = 1, b = –2, and c = 7. Substituting Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The check is left to the student. The solutions are The check is left to the student. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Approximating Solutions When the solution of an equation is irrational, a rational-number approximation is often useful. This is often the case in real-world applications similar to those found in section 8.3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use a calculator to approximate Solution Take the time to familiarize yourself with your calculator: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley