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Warm-up 1. Solve the following quadratic equation by Completing the Square: x 2 - 10x + 15 = 0 2. Convert the following quadratic equation to vertex format y = 2x 2 – 8x + 20

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Chapter 4 Section 4-8 The Discriminant

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Objectives I can calculate the value of the discriminant to determine the number and types of solutions to a quadratic equation.

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Quadratic Review Quadratic Equation in standard format: y = ax 2 + bx + c Solutions (roots) are where the graph crosses or touches the x-axis. Solutions can be real or imaginary

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Types of Solutions

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2 Real Solutions 1 Real Solution 2 Imaginary Solutions

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Key Concept for this Section What happens when you square any number like below: x 2 = ? It is always POSITIVE!! This is always the biggest mistake in this section

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Key Concept #2 What happens when you subtract a negative number like below: 3 - -4 = ? It becomes ADDITION!! This is 2 nd biggest error on this unit!

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The Quadratic Formula The solutions of any quadratic equation in the format ax 2 + bx + c = 0, where a 0, are given by the following formula: x = The quadratic equation must be set equal to ZERO before using this formula!!

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Discriminant The discriminant is just a part of the quadratic formula listed below: b 2 – 4ac The value of the discriminant determines the number and type of solutions.

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Discriminant Possibilities Value of b 2 -4ac Discriminant is a Perfect Square? # of Solutions Type of Solutions > 0Yes2Rational > 0No2Irrational < 02Imaginary = 01Rational

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Example 1 What are the nature of roots for the equation: x 2 – 8x + 16 = 0 a = 1, b = -8, c = 16 Discriminant: b 2 – 4ac (-8) 2 – 4(1)(16) 64 – 64 = 0 1 Rational Solution

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Example 2 What are the nature of roots for the equation: x 2 – 5x - 50 = 0 a = 1, b = -5, c = -50 Discriminant: b 2 – 4ac (-5) 2 – 4(1)(-50) 25 – (-200) = 225, which is a perfect square 2 Rational Solutions

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Example 3 What are the nature of roots for the equation: 2x 2 – 9x + 8 = 0 a = 2, b = -9, c = 8 Discriminant: b 2 – 4ac (-9) 2 – 4(2)(8) 81 – 64 = 17, which is not a perfect square 2 Irrational Solutions

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Example 4 What are the nature of roots for the equation: 5x 2 + 42= 0 a = 5, b = 0, c = 42 Discriminant: b 2 – 4ac (0) 2 – 4(5)(42) 0 – 840 = -840 2 Imaginary Imaginary

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GUIDED PRACTICE for Example 4 Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 4. 2x 2 + 4x – 4 = 0 SOLUTION Equation DiscriminantSolution(s) ax 2 + bx + c = 0b 2 – 4ac 2x 2 + 4x – 4 = 0 4 2 – 4(2)(– 4 ) x = – b+ b 2 – 4ac 2ac = 48 Two irrational solutions

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GUIDED PRACTICE for Example 4 5. SOLUTION Equation DiscriminantSolution(s) ax 2 + bx + c = 0b 2 – 4ac 12 2 – 4(12)(3 ) x = – b+ b 2 – 4ac 2ac = 0 One rational solution 3x 2 + 12x + 12 = 0

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6. SOLUTION Equation DiscriminantSolution(s) ax 2 + bx + c = 0b 2 – 4ac x = – b+ b 2 – 4ac 2ac GUIDED PRACTICE for Example 4 8x 2 = 9x – 11 8x 2 – 9x + 11 = 0(– 9) 2 – 4(8)(11 ) = – 271 Two imaginary solutions

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7. SOLUTION Equation DiscriminantSolution(s) ax 2 + bx + c = 0b 2 – 4ac x = – b+ b 2 – 4ac 2ac GUIDED PRACTICE for Example 4 7x 2 – 2x = 5 (– 2) 2 – 4(7)(– 5 ) = 144 Two rational solutions 7x 2 – 2x – 5 = 0

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Homework WS 7-2

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