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Good Morning! Please get the e-Instruction Remote with your assigned Calculator Number on it, and have a seat… Then answer this question by aiming the remote at the white receiver by my computer and pressing your answer choice!

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Warm Up Write each function in standard form. Evaluate b 2 – 4ac for the given values of the valuables. 1. f(x) = (x – 4) 2 + 3 2. g(x) = 2(x + 6) 2 – 11 f(x) = x 2 – 8x + 19 g(x) = 2x 2 + 24x + 61 4. a = 1, b = 3, c = –3 3. a = 2, b = 7, c = 5 9 21

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Solve quadratic equations using the Quadratic Formula. Objectives

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You have learned several methods for solving quadratic equations: graphing, making tables, factoring, using square roots, and completing the square. Another method is to use the Quadratic Formula, which allows you to solve a quadratic equation in standard form. By completing the square on the standard form of a quadratic equation, you can determine the Quadratic Formula.

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Make sure the equation is in standard form before you evaluate the discriminant, b 2 – 4ac. Caution!

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Find the type and number of solutions for the equation. Example 3A: Analyzing Quadratic Equations by Using the Discriminant x 2 + 36 = 12x x 2 – 12x + 36 = 0 b 2 – 4ac (–12) 2 – 4(1)(36) 144 – 144 = 0 b 2 – 4ac = 0 The equation has one distinct real solution.

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Find the type and number of solutions for the equation. Example 3B: Analyzing Quadratic Equations by Using the Discriminant x 2 + 40 = 12x x 2 – 12x + 40 = 0 b 2 – 4ac (–12) 2 – 4(1)(40) 144 – 160 = –16 b 2 –4ac < 0 The equation has two distinct nonreal complex solutions.

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Find the type and number of solutions for the equation. Example 3C: Analyzing Quadratic Equations by Using the Discriminant x 2 + 30 = 12x x 2 – 12x + 30 = 0 b 2 – 4ac (–12) 2 – 4(1)(30) 144 – 120 = 24 b 2 – 4ac > 0 The equation has two distinct real solutions.

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The discriminant is part of the Quadratic Formula that you can use to determine the number of real roots of a quadratic equation.

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You can use the Quadratic Formula to solve any quadratic equation that is written in standard form, including equations with real solutions or complex solutions.

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Quadratic Song Video!

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Khan Academy Video!

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Find the zeros of f(x)= 2x 2 – 16x + 27 using the Quadratic Formula. Example 1: Quadratic Functions with Real Zeros Set f(x) = 0. Write the Quadratic Formula. Substitute 2 for a, –16 for b, and 27 for c. Simplify. Write in simplest form. 2x 2 – 16x + 27 = 0

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Find the zeros of f(x) = x 2 + 3x – 7 using the Quadratic Formula. Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, 3 for b, and –7 for c. Simplify. Write in simplest form. Check It Out! Example 1a x 2 + 3x – 7 = 0

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Find the zeros of f(x)= x 2 – 8x + 10 using the Quadratic Formula. Set f(x) = 0. Write the Quadratic Formula. Substitute 1 for a, –8 for b, and 10 for c. Simplify. Write in simplest form. Check It Out! Example 1b x 2 – 8x + 10 = 0

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Find the zeros of f(x) = 4x 2 + 3x + 2 using the Quadratic Formula. Example 2: Quadratic Functions with Complex Zeros Set f(x) = 0. Write the Quadratic Formula. Substitute 4 for a, 3 for b, and 2 for c. Simplify. Write in terms of i. f(x)= 4x 2 + 3x + 2

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Find the zeros of g(x) = 3x 2 – x + 8 using the Quadratic Formula. Set f(x) = 0 Write the Quadratic Formula. Substitute 3 for a, –1 for b, and 8 for c. Simplify. Write in terms of i. Check It Out! Example 2

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The graph shows related functions. Notice that the number of real solutions for the equation can be changed by changing the value of the constant c.

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Properties of Solving Quadratic Equations

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No matter which method you use to solve a quadratic equation, you should get the same answer. Helpful Hint

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Lesson Quiz: Part I Find the zeros of each function by using the Quadratic Formula. 1. f(x) = 3x 2 – 6x – 5 2. g(x) = 2x 2 – 6x + 5 Find the type and member of solutions for each equation. 3. x 2 – 14x + 504. x 2 – 14x + 48 2 distinct nonreal complex 2 distinct real

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Homework! Holt Chapter 5 Section 6 Page 361 # 19-39 odd, 45-53 odd

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