1. Chapter 1.4
Quadratic Equations

2. Quadratic Equation in One Variable
An equation that can be written in the form
ax2 + bx + c = 0
where a, b, and c, are real numbers,
is a quadratic equation

3. A quadratic equation is a second-degree equation—that is, an equation with a squared term and no terms of greater degree.
x2 =25,
4x2 + 4x – 5 = 0,
3x2 = 4x - 8

4. A quadratic equation written in the form
ax2 + bx + c = 0 is in standard form.

5. This method depends on the zero-factor property.
Solving a Quadratic Equation
Factoring is the simplest method of solving a quadratic equation (but one not always easily applied).
This method depends on the zero-factor property.

6. Zero-Factor Property
If two numbers have a product of 0 then at least one of the numbers must be zero
If ab= then a = 0 or b = 0

7. Example 1. Using the zero factor property.
Solve 6x2 + 7x = 3

8. A quadratic equation of the form x2 = k can also be solved by factoring.

9. Square root property
If x2 = k, then

10. Example 2 Using the Square Root Property
Solve each quadratic equation.
x2 = 17

11. Example 2 Using the Square Root Property
Solve each quadratic equation.
x2 = -25

12. Example 2 Using the Square Root Property
Solve each quadratic equation.
(x-4)2 = 12

13. Completing the Square
Any quadratic equation can be solved by the method of completing the square.

14. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

15. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

16. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

17. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

18. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

19. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

20. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

21. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

22. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

23. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

24. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

25. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

26. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

27. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

28. Example 3 Using the Method of Completing the Square, a = 1
Solve x2 – 4x – 14 = 0

29. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

30. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

31. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

32. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

33. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

34. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

35. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

36. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

37. Example 4 Using the Method of Completing the Square, a ≠1
Solve 9x2 – 12x – 1 = 0

38. Example 4 Using the Method of Completing the Square, a ≠1

39. Example 4 Using the Method of Completing the Square, a ≠1

40. Example 4 Using the Method of Completing the Square, a ≠1

41. Example 4 Using the Method of Completing the Square, a ≠1

42. Example 4 Using the Method of Completing the Square, a ≠1

43. Example 4 Using the Method of Completing the Square, a ≠1

44. Example 4 Using the Method of Completing the Square, a ≠1

45. The Quadratic Formula
Watch the derivation

46. Example 5 Using the Quadratic Formula
(Real Solutions)
Solve x2 -4x = -2

47. Example 6 Using the Quadratic Formula
(Non-real Complex Solutions)
Solve 2x2 = x – 4

48. Example 7 Solving a Cubic Equation
Solve x3 + 8 = 0

49. Example 8 Solving a Variable That is Squared
Solve for the specified variable.

50. Example 8 Solving a Variable That is Squared
Solve for the specified variable.

51. b2 -4ac, is called the discriminant.
The Discriminant The quantity under the radical in the quadratic formula,
b2 -4ac, is called the discriminant.
Discriminant

52. Then the numbers a, b, and c are integers, the value of the discriminant can be used to determine whether the solution of a quadratic equation are rational, irrational, or nonreal complex numbers, as shown in the following table.

53. Discriminant Number of Solutions Kind of Solutions Positive Two
(Perfect Square)
Two
Rational
Positive
(but not a Perfect Square)
Two
Irrational
Zero
One
(a double solution)
Rational
Negative
Two
Nonreal complex

54. Example 9 Using the Discriminant
Determine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers.
5x2 + 2x – 4 = 0

55. Example 9 Using the Discriminant
Determine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers.
x2 – 10x = -25

56. Example 9 Using the Discriminant
Determine the number of solutions and tell whether they are rational, irrational, or nonreal complex numbers.
2x2 – x + 1 = 0

57. Homework 1.4 # 1-79