Chapter 9 Section 1

Solving Quadratic Equations by the Square Root Property 9.1 Solving Quadratic Equations by the Square Root Property Review the zero-factor property. Solve equations of the form x2 = k, where k > 0. Solve equations of the form (ax + b)2 = k, where k > 0. Use formulas involving squared variables. 2 3 4

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

Review the zero-factor property. Objective 1 Review the zero-factor property. Slide 9.1-4

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 x2 + 4x + 3 = 0 by factoring, using the zero-factor property. Standard Form Zero-Factor Property If a and b are real number and if ab = 0, then a = 0 or b = 0. Slide 9.1-5

Solve each equation by the zero-factor property. EXAMPLE 1 Solving Quadratic Equations by the Zero-Factor Property Solve each equation by the zero-factor property. 2x2 − 3x + 1 = 0 x2 = 25 Solution: Slide 9.1-6

Solve equations of the form x2 = k, where k > 0. Objective 2 Solve equations of the form x2 = k, where k > 0. Slide 9.1-7

Solve equations of the form x2 = k, where k > 0. We might also have solved x2 = 9 by noticing that x must be a number whose square is 9. Thus, or This can be generalized as the square root property. Square Root Property If k is a positive number and if x2 = 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. Slide 9.1-8

Solving Quadratic Equations of the Form x2 = k EXAMPLE 2 Solving Quadratic Equations of the Form x2 = k Solve each equation. Write radicals in simplified form. Solution: Slide 9.1-9

Solve equations of the form (ax + b)2 = k, where k > 0. Objective 3 Solve equations of the form (ax + b)2 = k, where k > 0. Slide 9.1-10

Solve equations of the form (ax + b)2 = k, where k > 0. 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. Slide 9.1-11

Solving Quadratic Equations of the Form (x + b)2 = k EXAMPLE 3 Solving Quadratic Equations of the Form (x + b)2 = k Solve (p – 4)2 = 3. Solution: Slide 9.1-12

Solving a Quadratic Equation of the Form (ax + b)2 = k EXAMPLE 4 Solving a Quadratic Equation of the Form (ax + b)2 = k Solve (5m + 1)2 = 7. Solution: Slide 9.1-13

EXAMPLE 5 Solve (7z + 1)2 = –1. Solution: Recognizing a Quadratic Equation with No Real Solutions Solve (7z + 1)2 = –1. Solution: Slide 9.1-14

Use formulas involving squared variables. Objective 4 Use formulas involving squared variables. Slide 9.1-15

Finding the Length of a Bass EXAMPLE 6 Finding the Length of a Bass 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 17.48 in. Slide 9.1-16

HOMEWORK (9.1) Book: Beginning Algebra Page: 557 Exercises 5 to 19 Page: 558 Exercises 37 to 47