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Hprec2 2

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Hprec2 2

1. 1. 2-2: Solving Quadratic Equations Algebraically © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Solve by factoring •Solve by taking square root of both sides •Solve by completing the square •Solve by using quadratic formula
2. 2. Definition A quadratic, or second degree equation is one that can be written in the form 2 0ax bx c+ + = for real constants a,b, and c with a≠0. This is the standard form for a quadratic equation.
3. 3. Important Idea There are 4 techniques to algebraically solve quadratic equations: •Factoring •Taking square root of both sides •Completing the square •Using quadratic formula
4. 4. Example Solve by factoring: 2 3 10x x− =
5. 5. Definition The zero product property: If the product of real numbers is zero, then one or both of the numbers must be zero
6. 6. Example Solve by factoring: What is wrong with this: 2 6 ( 1) 6 6 & 1 6 7 x x x x x x x − = − = = − = ⇒ =
7. 7. Try This Solve by factoring: 2 2 3 1 0t t+ + = 1 & 1 2 t t= − = − Can you think of a way to check your answer?
8. 8. Try This Solve by factoring: 2 18 23 6x x= + 2 3 or 9 2 x x= − = Hint: write in standard form
9. 9. Example Solve by taking the square root of both sides: 2 4 16x =a. 2 2 15x =b.
10. 10. Try This Solve by taking the square root of both sides: 2 4 16x = 2x = ±
11. 11. Try This Solve by taking the square root of both sides. Give exact and approximate solutions. 2 3 16x = 4 3 2.309 3 x = ± = ±
12. 12. Example Solve by taking the square root of both sides: What is wrong with this? 2 4x = −
13. 13. Example Complete the square for: 2 12x x+ 1. Half the coefficient of x: 1/2 of 12=6 2. Square this number and add to the expression 36+
14. 14. Important Idea Completing the square is the process of finding the number that will make the expression a perfect square trinomial.
15. 15. Important Idea 2 12 36x x+ + is a perfect square trinomial because it factors as: 2 ( 6)( 6) ( 6)x x x+ + = +
16. 16. Try This Complete the square for: 2 8x x+ then factor your result 2 2 8 16 ( 4)x x x+ + = +
17. 17. Example Complete the square for: 2 3 4 y y+ then factor your result. Use fractions only.
18. 18. Example Solve by completing the square: 2 8 14 0x x+ + = 1. Move the constant to the right: 2 8 14x x+ = −
19. 19. Example Solve by completing the square: 2 8 14 0x x+ + = 2. Complete the square and add to the left and right: 2 8 16 14 16x x+ + = − +
20. 20. Example 2 ( 4) 2x + = 4 2x + = ± Solve by completing the square: 2 8 14 0x x+ + = 3. Factor left side and solve: 4 2x = − ±
21. 21. Try This Solve by completing the square: 2 4 1 0x x− + = 2 3x = ±
22. 22. Example Solve by completing the square… 2 6 2 0x x− − = Before you complete the square, the coefficient of the squared term must be 1
23. 23. Try This Solve by completing the square…fractions only. 2 2 13 15 0x x+ + = 3 , 5 2 x = − −
24. 24. Definition The solutions to 2 0ax bx c+ + = are: 2 4 2 b b ac x a − ± − = These solutions are called the Quadratic Formula
25. 25. Important Idea 2 4 2 b b ac x a − ± − = 2 4b ac− is called the discriminant 1. If 2 4 0b ac− > there are 2 real solutions
26. 26. Important Idea 2 4 2 b b ac x a − ± − = 2 4b ac− is called the discriminant 2. If 2 4 0b ac− = there is 1 real solution
27. 27. Important Idea 2 4 2 b b ac x a − ± − = 2 4b ac− is called the discriminant 3. If 2 4 0b ac− < there are no real solutions
28. 28. Example Solve using the quadratic formula. Leave answer in simplified radical form. 2 2 1x x= +
29. 29. Try This Solve using the quadratic formula. Leave answer in simplified radical form. 2 4 3 5x x− = 3 89 8 x ± =
30. 30. Example Solve using the quadratic formula. Leave answer in simplified radical form. 4 2 4 13 3 0x x− + =
31. 31. Lesson Close State the quadratic formula from memory.