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Warm-up

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. Chapter 4. Section 4-8 The Discriminant. Objectives.

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Warm-up

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  1. Warm-up • 1. Solve the following quadratic equation by Completing the Square: • x2 - 10x + 15 = 0 • 2. Convert the following quadratic equation to vertex format • y = 2x2 – 8x + 20

  2. Chapter 4 Section 4-8 The Discriminant

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

  4. 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

  5. Types of Solutions

  6. Types of Solutions 2 Real Solutions 1 Real Solution 2 Imaginary Solutions

  7. 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

  8. 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!

  9. 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: • x = The quadratic equation must be set equal to ZERO before using this formula!!

  10. Discriminant • The discriminant is just a part of the quadratic formula listed below: b2 – 4ac • The value of the discriminant determines the number and type of solutions.

  11. Discriminant Possibilities

  12. 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

  13. 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

  14. 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

  15. 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

  16. – b+b2– 4ac x = 2ac for Example 4 GUIDED PRACTICE Find the discriminant of the quadratic equation and give the number and type of solutions of the equation. 2x2 + 4x – 4 = 0 SOLUTION Discriminant Solution(s) Equation ax2 + bx + c = 0 b2 – 4ac 2x2 + 4x – 4 = 0 42 – 4(2)(– 4 ) Two irrational solutions = 48

  17. – b+b2– 4ac x = 2ac for Example 4 GUIDED PRACTICE 3x2 + 12x + 12 = 0 SOLUTION Discriminant Solution(s) Equation ax2 + bx + c = 0 b2 – 4ac 3x2 + 12x + 12 = 0 122 – 4(12)(3 ) = 0 One rational solution

  18. – b+b2– 4ac x = 2ac for Example 4 GUIDED PRACTICE 8x2 = 9x – 11 SOLUTION Discriminant Solution(s) Equation ax2 + bx + c = 0 b2 – 4ac 8x2 – 9x + 11 = 0 (– 9)2 – 4(8)(11 ) = – 271 Two imaginary solutions

  19. – b+b2– 4ac x = 2ac for Example 4 GUIDED PRACTICE 7x2 – 2x = 5 SOLUTION Discriminant Solution(s) Equation ax2 + bx + c = 0 b2 – 4ac 7x2 – 2x –5= 0 (– 2)2 – 4(7)(– 5 ) = 144 Two rational solutions

  20. Homework • WS 7-2

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