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11.1 Solving Quadratic Equations by the Square Root Property

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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 11.1- 2 Solving Quadratic Equations by the Square Root Property

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Review the zero-factor property. Objective 1 Slide 11.1- 3

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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 11.1- 4 Review the zero-factor property.

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Solve each equation by the zero-factor property. 2x 2 − 3x + 1 = 0x 2 = 25 Solution: Slide 11.1-6 Solving Quadratic Equations by the Zero-Factor Property CLASSROOM EXAMPLE 1

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Objective 2 Solve equations of the form x 2 = k, where k > 0. Slide 11.1-7

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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 11.1-8 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.

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Solve each equation. Write radicals in simplified form. Solution: Slide 11.1-9 Solving Quadratic Equations of the Form x 2 = k CLASSROOM EXAMPLE 2

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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 9.1- 9 CLASSROOM EXAMPLE 3 Using the Square Root Property in an Application Solution:

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Objective 3 Solve equations of the form (ax + b) 2 = k, where k > 0. Slide 11.1-11

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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 11.1-12

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Solve (p – 4) 2 = 3. Solution: Slide 11.1-13 Solving Quadratic Equations of the Form (x + b) 2 = k CLASSROOM EXAMPLE 4

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Solve (5m + 1) 2 = 7. Solution: Slide 11.1-14 Solving a Quadratic Equation of the Form (ax + b) 2 = k CLASSROOM EXAMPLE 5

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Solve quadratic equations with solutions that are not real numbers. Objective 4 Slide 11.1- 14

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Solve the equation. The solution set is Slide 11.1- 15 CLASSROOM EXAMPLE 6 Solve for Nonreal Complex Solutions Solution:

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The solution set is Slide 11.1- 16 CLASSROOM EXAMPLE 6 Solve for Nonreal Complex Solutions (cont’d) Solve the equation. Solution:

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The solution set is Slide 11.1- 17 CLASSROOM EXAMPLE 6 Solve for Nonreal Complex Solutions (cont’d) Solve the equation. Solution:

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4.2 Integer Exponents and the Quotient Rule

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