Analyzing Solutions to Quadratic Equations

1. Section 6-10
Analyzing Solutions to Quadratic
Equations
Tuesday, February 17, 2009

2. Warm-up
Tuesday, February 17, 2009

3. Warm-up
In-Class Activity on p. 399
Tuesday, February 17, 2009

4. Warm-up
In-Class Activity on p. 399
Work with a partner or two
Tuesday, February 17, 2009

5. Warm-up
In-Class Activity on p. 399
Work with a partner or two
Use your graphing calculators
Tuesday, February 17, 2009

6. Warm-up
In-Class Activity on p. 399
Work with a partner or two
Use your graphing calculators
Fill out the table
Tuesday, February 17, 2009

7. Ways to find real
solutions to quadratics
Tuesday, February 17, 2009

8. Ways to find real
solutions to quadratics
1. Graphing
Tuesday, February 17, 2009

9. Ways to find real
solutions to quadratics
1. Graphing
2. Systems (Quadratic in standard form
and y = 0
Tuesday, February 17, 2009

10. Ways to find real
solutions to quadratics
1. Graphing
2. Systems (Quadratic in standard form
and y = 0
3. Quadratic Formula
Tuesday, February 17, 2009

11. Ways to find real
solutions to quadratics
1. Graphing
2. Systems (Quadratic in standard form
and y = 0
3. Quadratic Formula
4. Factoring...Chapter 11
Tuesday, February 17, 2009

12. Discriminant
Tuesday, February 17, 2009

13. Discriminant
2
b − 4ac
Tuesday, February 17, 2009

14. Discriminant
2
b − 4ac
From the quadratic formula
(the “stuff” in the radical)
Tuesday, February 17, 2009

15. Discriminant
2
b − 4ac
From the quadratic formula
(the “stuff” in the radical)
Tells how many solutions
there will be for a quadratic
Tuesday, February 17, 2009

16. Three different possibilities
for the discriminant
When using a quadratic in standard form
2
ax + bx + c = 0
Tuesday, February 17, 2009

17. Three different possibilities
for the discriminant
When using a quadratic in standard form
2
ax + bx + c = 0
2
1. b − 4ac > 0
Tuesday, February 17, 2009

18. Three different possibilities
for the discriminant
When using a quadratic in standard form
2
ax + bx + c = 0
2
2 real solutions
1. b − 4ac > 0
Tuesday, February 17, 2009

19. Three different possibilities
for the discriminant
When using a quadratic in standard form
2
ax + bx + c = 0
2
2 real solutions
1. b − 4ac > 0
2
2. b − 4ac = 0
Tuesday, February 17, 2009

20. Three different possibilities
for the discriminant
When using a quadratic in standard form
2
ax + bx + c = 0
2
2 real solutions
1. b − 4ac > 0
2
2. b − 4ac = 0 1 real solution
Tuesday, February 17, 2009

21. Three different possibilities
for the discriminant
When using a quadratic in standard form
2
ax + bx + c = 0
2
2 real solutions
1. b − 4ac > 0
2
2. b − 4ac = 0 1 real solution
2
3. b − 4ac < 0
Tuesday, February 17, 2009

22. Three different possibilities
for the discriminant
When using a quadratic in standard form
2
ax + bx + c = 0
2
2 real solutions
1. b − 4ac > 0
2
2. b − 4ac = 0 1 real solution
2
3. b − 4ac < 0 2 complex solutions
Tuesday, February 17, 2009

23. Example 1
Determine the number of roots (solutions), then
solve.
2
a. 6x − 3x − 4 = 0
Tuesday, February 17, 2009

24. Example 1
Determine the number of roots (solutions), then
solve.
2
a. 6x − 3x − 4 = 0
2
b − 4ac
Tuesday, February 17, 2009

25. Example 1
Determine the number of roots (solutions), then
solve.
2
a. 6x − 3x − 4 = 0
2
b − 4ac
2
= (−3) − 4(6)(−4)
Tuesday, February 17, 2009

26. Example 1
Determine the number of roots (solutions), then
solve.
2
a. 6x − 3x − 4 = 0
2
b − 4ac
2
= (−3) − 4(6)(−4)
= 9 + 96
Tuesday, February 17, 2009

27. Example 1
Determine the number of roots (solutions), then
solve.
2
a. 6x − 3x − 4 = 0
2
b − 4ac
2
= (−3) − 4(6)(−4)
= 9 + 96
= 105
Tuesday, February 17, 2009

28. Example 1
Determine the number of roots (solutions), then
solve.
2
a. 6x − 3x − 4 = 0
2
b − 4ac
2
= (−3) − 4(6)(−4)
= 9 + 96
= 105
2 real solutions
Tuesday, February 17, 2009

29. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
a. 6x − 3x − 4 = 0 x=
2a
2
b − 4ac
2
= (−3) − 4(6)(−4)
= 9 + 96
= 105
2 real solutions
Tuesday, February 17, 2009

30. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
a. 6x − 3x − 4 = 0 x=
2a
2
b − 4ac
3 ± 105
=
2
= (−3) − 4(6)(−4) 12
= 9 + 96
= 105
2 real solutions
Tuesday, February 17, 2009

31. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
a. 6x − 3x − 4 = 0 x=
2a
2
b − 4ac
3 ± 105
=
2
= (−3) − 4(6)(−4) 12
= 9 + 96
3 + 105 3 − 105
= =
= 105 12 12
2 real solutions
Tuesday, February 17, 2009

32. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
a. 6x − 3x − 4 = 0 x=
2a
2
b − 4ac
3 ± 105
=
2
= (−3) − 4(6)(−4) 12
= 9 + 96
3 + 105 3 − 105
= =
= 105 12 12
≈ 1.1
2 real solutions
Tuesday, February 17, 2009

33. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
a. 6x − 3x − 4 = 0 x=
2a
2
b − 4ac
3 ± 105
=
2
= (−3) − 4(6)(−4) 12
= 9 + 96
3 + 105 3 − 105
= =
= 105 12 12
≈ 1.1
2 real solutions ≈ −.6
Tuesday, February 17, 2009

34. Example 1
Determine the number of roots (solutions), then
solve.
2
b. 7x + 2x + 7 = 0
Tuesday, February 17, 2009

35. Example 1
Determine the number of roots (solutions), then
solve.
2
b. 7x + 2x + 7 = 0
2
b − 4ac
Tuesday, February 17, 2009

36. Example 1
Determine the number of roots (solutions), then
solve.
2
b. 7x + 2x + 7 = 0
2
b − 4ac
2
= 2 − 4(7)(7)
Tuesday, February 17, 2009

37. Example 1
Determine the number of roots (solutions), then
solve.
2
b. 7x + 2x + 7 = 0
2
b − 4ac
2
= 2 − 4(7)(7)
= 4 − 196
Tuesday, February 17, 2009

38. Example 1
Determine the number of roots (solutions), then
solve.
2
b. 7x + 2x + 7 = 0
2
b − 4ac
2
= 2 − 4(7)(7)
= 4 − 196
= −192
Tuesday, February 17, 2009

39. Example 1
Determine the number of roots (solutions), then
solve.
2
b. 7x + 2x + 7 = 0
2
b − 4ac
2
= 2 − 4(7)(7)
= 4 − 196
= −192
2 complex solutions
Tuesday, February 17, 2009

40. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
b. 7x + 2x + 7 = 0 x=
2a
2
b − 4ac
2
= 2 − 4(7)(7)
= 4 − 196
= −192
2 complex solutions
Tuesday, February 17, 2009

41. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
b. 7x + 2x + 7 = 0 x=
2a
2
b − 4ac
−2 ± −192
2
= 2 − 4(7)(7) =
14
= 4 − 196
= −192
2 complex solutions
Tuesday, February 17, 2009

42. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
b. 7x + 2x + 7 = 0 x=
2a
2
b − 4ac
−2 ± i 192
−2 ± −192
2
= 2 − 4(7)(7) = =
14 14
= 4 − 196
= −192
2 complex solutions
Tuesday, February 17, 2009

43. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
b. 7x + 2x + 7 = 0 x=
2a
2
b − 4ac
−2 ± i 192
−2 ± −192
2
= 2 − 4(7)(7) = =
14 14
= 4 − 196 −2 + i 192 −2 − i 192
= =
14 14
= −192
2 complex solutions
Tuesday, February 17, 2009

44. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
b. 7x + 2x + 7 = 0 x=
2a
2
b − 4ac
−2 ± i 192
−2 ± −192
2
= 2 − 4(7)(7) = =
14 14
= 4 − 196
−2 + i 192 −2 − i 192
= =
14 14
= −192
1 43
2 complex solutions = − + i
7 7
Tuesday, February 17, 2009

45. Example 1
Determine the number of roots (solutions), then
solve.
2
−b ± b − 4ac
2
b. 7x + 2x + 7 = 0 x=
2a
2
b − 4ac
−2 ± i 192
−2 ± −192
2
= 2 − 4(7)(7) = =
14 14
= 4 − 196
−2 + i 192 −2 − i 192
= =
14 14
= −192
1 43 1 43
2 complex solutions = − + i =− − i
7 7 7 7
Tuesday, February 17, 2009

46. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
Tuesday, February 17, 2009

47. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
2
205 = −.005x + 2x + 3.5
Tuesday, February 17, 2009

48. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
2
205 = −.005x + 2x + 3.5
2
0 = −.005x + 2x − 201.5
Tuesday, February 17, 2009

49. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
2
205 = −.005x + 2x + 3.5
2
0 = −.005x + 2x − 201.5
2
b − 4ac
Tuesday, February 17, 2009

50. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
2
205 = −.005x + 2x + 3.5
2
0 = −.005x + 2x − 201.5
2
b − 4ac
2
= 2 − 4(−.005)(−201.5)
Tuesday, February 17, 2009

51. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
2
205 = −.005x + 2x + 3.5
2
0 = −.005x + 2x − 201.5
2
b − 4ac
2
= 2 − 4(−.005)(−201.5)
= 4 − 4.03
Tuesday, February 17, 2009

52. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
2
205 = −.005x + 2x + 3.5
2
0 = −.005x + 2x − 201.5
2
b − 4ac
2
= 2 − 4(−.005)(−201.5)
= 4 − 4.03
= −.03
Tuesday, February 17, 2009

53. Example 2
2
The function h(x ) = −.005x + 2x + 3.5 gives the
height h(x ) and distance from home plate x for a
ball hit by Pop Fligh. Does Pop’s ball reach a height of
205 ft?
2
205 = −.005x + 2x + 3.5
2
0 = −.005x + 2x − 201.5
2
b − 4ac
2
= 2 − 4(−.005)(−201.5)
= 4 − 4.03
= −.03
The ball does not reach a height of 205 ft.
Tuesday, February 17, 2009

54. Homework
Tuesday, February 17, 2009

55. Homework
p. 405 #1 - 28 (skip 16, 17)
“Never let the fear of failure be an excuse for not
trying. Society tells us that to fail is the most terrible
thing in the world, but I know it isn’t. Failure is part
of what makes us human.” - Amber Deckers
Tuesday, February 17, 2009