1. The Quadratic Formula and the Discriminant
Algebra II Mr. Gilbert
Chapter 6.5
The Quadratic Formula and the Discriminant
Standard & Honors
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2. Agenda Warm up Homework Review Lesson New Homework Check your answers
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3. Click the mouse button or press the Space Bar to display the answers.
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Click the mouse button or press the Space Bar to display the answers.
Transparency 5

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5. Homework Review
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6. Example 1 Two Rational Roots (4) Example 2 One Rational Root (3)
Example 3 Irrational Roots (4)
Example 4 Complex Roots (5)
Example 5 Describe Roots
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Lesson 5 Contents

7. Solve by using the Quadratic Formula.
First, write the equation in the form and identify a, b, and c.
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Example 5-1a

8. Then, substitute these values into the Quadratic Formula.
Replace a with 1, b with –8, and c with –33.
Simplify.
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Example 5-1a

9. Answer: The solutions are 11 and –3.
Simplify.
Write as two equations. or
Simplify.
Answer: The solutions are 11 and –3.
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Example 5-1a

10. Solve by using the Quadratic Formula.
Answer: 2, –15
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Example 5-1b

11. Solve by using the Quadratic Formula.
Identify a, b, and c. Then, substitute these values into the Quadratic Formula.
Quadratic Formula
Replace a with 1, b with –34, and c with 289.
Simplify.
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Example 5-2a

12. Answer: The solution is 17.
Check A graph of the related function shows that there is one solution at
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Example 5-2a

13. Solve by using the Quadratic Formula.
Answer: 11
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Example 5-2b

14. Solve by using the Quadratic Formula.
Replace a with 1, b with –6, and c with 2.
Simplify.
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Example 5-3a

15. or
Answer: The exact solutions are and The approximate solutions are 0.4 and 5.6.
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Example 5-3a

16. Check. Check these results by graphing the related quadratic function,
Check Check these results by graphing the related quadratic function, Using the ZERO function of a graphing calculator, the approximate zeros of the related function are –2.9 and 0.9.
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Example 5-3a

17. Solve by using the Quadratic Formula.
Answer: or approximately 0.7 and 4.3
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Example 5-3b

18. Solve by using the Quadratic Formula.
Write the equation in the form
Now use the Quadratic Formula.
Quadratic Formula
Replace a with 1, b with –6, and c with 13.
Simplify.
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Example 5-4a

19. Answer: The solutions are the complex numbers and
Simplify.
Answer: The solutions are the complex numbers and
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Example 5-4a

20. A graph of the function shows that the solutions are complex, but it cannot help you find them.
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Example 5-4a

21. Sum of a square; Distributive Property
Check To check complex solutions, you must substitute them into the original equations. The check for is shown below.
Original equation
Sum of a square; Distributive Property
Simplify.
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Example 5-4a

22. Solve by using the Quadratic Formula.
Answer:
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Example 5-4b

23. Answer: The discriminant is 0, so there is one rational root.
Find the value of the discriminant for . Then describe the number and type of roots for the equation.
Answer: The discriminant is 0, so there is one rational root.
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Example 5-5a

24. Answer: The discriminant is negative, so there are two complex roots.
Find the value of the discriminant for . Then describe the number and type of roots for the equation.
Answer: The discriminant is negative, so there are two complex roots.
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Example 5-5a

25. Find the value of the discriminant for
Find the value of the discriminant for . Then describe the number and type of roots for the equation.
Answer: The discriminant is 80, which is not a perfect square. Therefore, there are two irrational roots.
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Example 5-5a

26. Find the value of the discriminant for
Find the value of the discriminant for . Then describe the number and type of roots for the equation.
Answer: The discriminant is 81, which is a perfect square. Therefore, there are two rational roots.
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Example 5-5a

27. Answer: 0; 1 rational root
Find the value of the discriminant for each quadratic equation. Then describe the number and type of roots for the equation. a. b. c. d.
Answer: 0; 1 rational root
Answer: –24; 2 complex roots
Answer: 5; 2 irrational roots
Answer: 64; 2 rational roots
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Example 5-5b

28. Homework - Honors See Syllabus 6.5
Pg multiples of 3, 40-45
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29. Homework
See Syllabus 6.5
Pg multiples of 3
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