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Chapter 1.4

Chapter 1.4. Quadratic Equations. Quadratic Equation in One Variable. An equation that can be written in the form ax 2 + bx + c = 0 where a, b, and c, are real numbers, is a quadratic equation.

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Chapter 1.4

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  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. 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= 0 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. x2 = k x2 – k=0

  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.

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