Chapter 9 Section 1

Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. 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. Use formulas involving squared variables

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Quadratic Equations by the Square Root Property In Section 6.5, we solved quadratic equations by factoring. Since not all quadratic equations can easily be solved by factoring, we must develop other methods. Slide 9.1-3

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Review the zero-factor property. Slide 9.1-4

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Quadratic Equations by the Square Root Property Recall that a quadratic equation is an equation that can be written in the form for real numbers a, b, and c, with a ≠ 0. We can solve the quadratic equation x 2 + 4x + 3 = 0 by factoring, using the zero-factor property. Slide Zero-Factor Property If a and b are real number and if ab = 0, then a = 0 or b = 0. Standard Form

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

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Solve equations of the form x 2 = k, where k > 0. Slide 9.1-7

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 EXAMPLE 2 Solving Quadratic Equations of the Form x 2 = k

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 EXAMPLE 3 Solving Quadratic Equations of the Form (x + b) 2 = k

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve (5m + 1) 2 = 7. Solution: Slide EXAMPLE 4 Solving a Quadratic Equation of the Form (ax + b) 2 = k

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve (7z + 1) 2 = –1. Solution: Slide EXAMPLE 5 Recognizing a Quadratic Equation with No Real Solutions

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 4 Use formulas involving squared variables. Slide

Copyright © 2012, 2008, 2004 Pearson Education, Inc. Use the formula, to approximate the length of a bass weighing 2.80 lb and having girth 11 in. Solution : The length of the bass is approximately in. Slide EXAMPLE 6 Finding the Length of a Bass