Quadratic Equations, Inequalities, and Functions Chapter 10 Quadratic Equations, Inequalities, and Functions

10.2 The Quadratic Formula

Derive the quadratic formula. Objectives Derive the quadratic formula. Solve quadratic equations using the quadratic formula. Use the discriminant to determine the number and type of solutions. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Derive the Quadratic Formula We can use the method of completing the square to solve any quadratic equation. However, completing the square can become tedious and time consuming. In this section, we begin with the general quadratic equation By applying the method of completing the square to this general equation, we can develop a formula for finding the solution of any specific quadratic equation. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Derive the Quadratic Formula Applying the Method of Completing the Square Copyright © 2010 Pearson Education, Inc. All rights reserved.

Derive the Quadratic Formula Applying the Method of Completing the Square Cont’d. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Derive the Quadratic Formula Applying the Method of Completing the Square Cont’d. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Stating and Applying the Quadratic Formula Copyright © 2010 Pearson Education, Inc. All rights reserved.

Stating and Applying the Quadratic Formula Copyright © 2010 Pearson Education, Inc. All rights reserved.

Statement of the Quadratic Formula You should always check each solution in the original equation. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Statement of the Quadratic Formula Always try factoring first. If factoring is difficult or impossible, use the quadratic equation. Copyright © 2010 Pearson Education, Inc. All rights reserved.

Using Quadratic Formula (Irrational Solutions) 10.2 The Quadratic Formula Using Quadratic Formula (Irrational Solutions) Copyright © 2010 Pearson Education, Inc. All rights reserved.

Using Quadratic Formula (Irrational Solutions) 10.2 The Quadratic Formula Using Quadratic Formula (Irrational Solutions) Copyright © 2010 Pearson Education, Inc. All rights reserved.

Simplifying the Results Using the Quadratic Formula Be sure to factor first; then divide out the common factor. Copyright © 2010 Pearson Education, Inc. All rights reserved.

10.2 The Quadratic Formula Using Quadratic Formula (Nonreal Complex Solutions) Copyright © 2010 Pearson Education, Inc. All rights reserved.

10.2 The Quadratic Formula Using Quadratic Formula (Nonreal Complex Solutions) Copyright © 2010 Pearson Education, Inc. All rights reserved.

10.2 The Quadratic Formula Using the Discriminant to Predict Number and Type of Solutions Discriminant Copyright © 2010 Pearson Education, Inc. All rights reserved.

What Information the Discriminant Predicts 10.2 The Quadratic Formula What Information the Discriminant Predicts If the discriminant is a perfect square (including 0), then the equation can be factored; otherwise, the quadratic formula should be used. Copyright © 2010 Pearson Education, Inc. All rights reserved.

What Information the Discriminant Predicts 10.2 The Quadratic Formula What Information the Discriminant Predicts The value of the discriminant is –44 and the equation has two nonreal complex solutions and is best solved with the quadratic formula. Because the discriminant is 0, there is only one rational solution, and the equation can be solved by factoring. Copyright © 2010 Pearson Education, Inc. All rights reserved.

What Information the Discriminant Predicts 10.2 The Quadratic Formula What Information the Discriminant Predicts The discriminant is a perfect square so there will be two rational solutions and the equation can be solved by factoring. The value of the discriminant is 109 and the equation has two irrational solutions and is best solved with the quadratic formula. Copyright © 2010 Pearson Education, Inc. All rights reserved.