Section 1 Chapter 11

Copyright © 2012, 2008, 2004 Pearson Education, Inc. 1 Objectives Solving Quadratic Equations by the Square Root Property Review the zero-factor property. Solve equations of the form x 2 = k, where k > 0. Solve equations of the form (ax + b) 2 = k, where k > 0. Solve quadratic equations with solutions that are not real numbers. 11.1

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Quadratic Equation An equation that can be written in the form where a, b, and c are real numbers, with a ≠ 0, is a quadratic equation. The given form is called standard form. Slide Solving Quadratic Equations by the Square Root Property

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Review the zero-factor property. Objective 1 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Zero-Factor Property If two numbers have a product of 0, then at least one of the numbers must be 0. That is, if ab = 0, then a = 0 or b = 0. Slide Review the zero-factor property.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve each equation by the zero-factor property. 2x 2 − 3x + 1 = 0x 2 = 25 Solution: Slide Solving Quadratic Equations by the Zero-Factor Property CLASSROOM EXAMPLE 1

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Solve equations of the form x 2 = k, where k > 0. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve equations of the form x 2 = k, where k > 0. We might also have solved x 2 = 9 by noticing that x must be a number whose square is 9. Thus, or This can be generalized as the square root property. Slide Square Root Property If k is a positive number and if x 2 = k, then or The solution set is which can be written (± is read “positive or negative” or “plus or minus.”) When we solve an equation, we must find all values of the variable that satisfy the equation. Therefore, we want both the positive and negative square roots of k.

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve each equation. Write radicals in simplified form. Solution: Slide Solving Quadratic Equations of the Form x 2 = k CLASSROOM EXAMPLE 2

Copyright © 2012, 2008, 2004 Pearson Education, Inc. An expert marksman can hold a silver dollar at forehead level, drop it, draw his gun, and shoot the coin as it passes waist level. If the coin falls about 4 ft, use the formula d = 16t 2 to find the time that elapses between the dropping of the coin and the shot. d = 16t 2 4 = 16t 2 By the square root property, Since time cannot be negative, we discard the negative solution. Therefore, 0.5 sec elapses between the dropping of the coin and the shot. Slide CLASSROOM EXAMPLE 3 Using the Square Root Property in an Application Solution:

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 3 Solve equations of the form (ax + b) 2 = k, where k > 0. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. In each equation in Example 2, the exponent 2 appeared with a single variable as its base. We can extend the square root property to solve equations in which the base is a binomial. Solve equations of the form (ax + b) 2 = k, where k > 0. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve (p – 4) 2 = 3. Solution: Slide Solving Quadratic Equations of the Form (x + b) 2 = k CLASSROOM EXAMPLE 4

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve (5m + 1) 2 = 7. Solution: Slide Solving a Quadratic Equation of the Form (ax + b) 2 = k CLASSROOM EXAMPLE 5

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve quadratic equations with solutions that are not real numbers. Objective 4 Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve the equation. The solution set is Slide CLASSROOM EXAMPLE 6 Solve for Nonreal Complex Solutions Solution:

Copyright © 2012, 2008, 2004 Pearson Education, Inc. The solution set is Slide CLASSROOM EXAMPLE 6 Solve for Nonreal Complex Solutions (cont’d) Solve the equation. Solution: