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Warm-up • 1. Solve the following quadratic equation by Completing the Square: • x2 - 10x + 15 = 0 x 5 10 • 2. Convert the following quadratic equation to vertex format • y = 2x2 – 8x + 20 y 2( x 2) 12 2 Chapter 4 Section 4-8 The Discriminant Objectives • I can calculate the value of the discriminant to determine the number and types of solutions to a quadratic equation. Quadratic Review • Quadratic Equation in standard format: • y = ax2 + bx + c • Solutions (roots) are where the graph crosses or touches the x-axis. • Solutions can be real or imaginary Types of Solutions Complex Number System Real Numbers Imaginary Numbers a bi Rational Irrational Types of Solutions 2 Real Solutions 1 Real Solution 2 Imaginary Solutions b b 4ac x 2a 2 2 What value of b -4ac gives each solution type? Key Concept for this Section • What happens when you square any number like below: • x2 = ? • It is always POSITIVE!! • This is always the biggest mistake in this section Key Concept #2 • What happens when you subtract a negative number like below: • 3 - -4 = ? • It becomes ADDITION!! • This is 2nd biggest error on this unit! The Quadratic Formula • The solutions of any quadratic equation in the format ax2 + bx + c = 0, where a 0, are given by the following formula: 2 b b 4ac • x= 2a The quadratic equation must be set equal to ZERO before using this formula!! Discriminant • The discriminant is just a part of the quadratic formula listed below: 2 b – 4ac • The value of the discriminant determines the number and type of solutions. Discriminant Possibilities Value of b2-4ac >0 >0 Discriminant # of is a Perfect Solutions Square? Yes 2 No Type of Solutions Rational 2 Irrational <0 2 Imaginary =0 1 Rational Example 1 • • • • • • • What are the nature of roots for the equation: x2 – 8x + 16 = 0 a = 1, b = -8, c = 16 Discriminant: b2 – 4ac (-8)2 – 4(1)(16) 64 – 64 = 0 1 Rational Solution Example 2 • • • • • • • What are the nature of roots for the equation: x2 – 5x - 50 = 0 a = 1, b = -5, c = -50 Discriminant: b2 – 4ac (-5)2 – 4(1)(-50) 25 – (-200) = 225, which is a perfect square 2 Rational Solutions Example 3 • • • • • • • What are the nature of roots for the equation: 2x2 – 9x + 8 = 0 a = 2, b = -9, c = 8 Discriminant: b2 – 4ac (-9)2 – 4(2)(8) 81 – 64 = 17, which is not a perfect square 2 Irrational Solutions Example 4 • • • • • • • What are the nature of roots for the equation: 5x2 + 42= 0 a = 5, b = 0, c = 42 Discriminant: b2 – 4ac (0)2 – 4(5)(42) 0 – 840 = -840 2 Imaginary Imaginary GUIDED PRACTICE for Example 4 Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 4. 2x2 + 4x – 4 = 0 SOLUTION Equation Discriminant ax2 + bx + c = 0 b2 – 4ac 2x2 + 4x – 4 = 0 42 – 4(2)(– 4 ) Solution(s) 2 x = – b+ b – 4ac 2ac Two irrational solutions = 48 for Example 4 GUIDED PRACTICE 5. 3x2 + 12x + 12 = 0 SOLUTION Equation ax2 + bx + c = 0 3x2 + 12x + 12 = 0 Discriminant b2 – 4ac 122 – 4(12)(3 ) Solution(s) 2 x = – b+ b – 4ac 2ac =0 One rational solution GUIDED PRACTICE 6. for Example 4 8x2 = 9x – 11 SOLUTION Equation ax2 + bx + c = 0 8x2 – 9x + 11 = 0 Discriminant b2 – 4ac (– 9)2 – 4(8)(11 ) Solution(s) 2 x = – b+ b – 4ac 2ac = – 271 Two imaginary solutions GUIDED PRACTICE 7. for Example 4 7x2 – 2x = 5 SOLUTION Equation ax2 + bx + c = 0 7x2 – 2x – 5 = 0 Discriminant b2 – 4ac (– 2)2 – 4(7)(– 5 ) Solution(s) 2 x = – b+ b – 4ac 2ac = 144 Two rational solutions Homework • WS 7-2