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6.5 – Solving Equations with Quadratic Techniques

6.5 – Solving Equations with Quadratic Techniques. Quadratic equations are in the form: a x 2 + b x + c ,. Quadratic equations are in the form: a x 2 + b x + c , where a, b, & c are integers. Quadratic equations are in the form: a x 2 + b x + c , where a, b, & c are integers exs. .

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6.5 – Solving Equations with Quadratic Techniques

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  1. 6.5 – Solving Equations with Quadratic Techniques

  2. Quadratic equations are in the form: ax2 + bx + c,

  3. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers

  4. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs.

  5. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs. • x2 + 5x + 2

  6. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs. • x2 + 5x + 2 • 2x2 – 18x + 13

  7. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs. • x2 + 5x + 2 • 2x2 – 18x + 13 • x2 – 9

  8. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs. • x2 + 5x + 2 • 2x2 – 18x + 13 • x2 – 9 x2+ 0x – 9

  9. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs. • x2 + 5x + 2 • 2x2 – 18x + 13 • x2 – 9 x2+ 0x – 9 • 2x2 + 8x

  10. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs. • x2 + 5x + 2 • 2x2 – 18x + 13 • x2 – 9 x2+ 0x – 9 • 2x2 + 8x 2x2 + 8x+ 0

  11. Quadratic equations are in the form: ax2 + bx + c, where a, b, & c are integers exs. • x2 + 5x + 2 • 2x2 – 18x + 13 • x2 – 9 x2+ 0x – 9 • 2x2 + 8x 2x2 + 8x+ 0 NOTE: Must have the “x2” term to be a quadratic!

  12. Ex. 1 Write each expression in quadratic form, if possible.

  13. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36

  14. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2

  15. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2)

  16. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

  17. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36

  18. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625

  19. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625 (4x3)2

  20. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625 (4x3)2 – 625

  21. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625 (4x3)2 – 625 c. x½– 9x¼ + 16

  22. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625 (4x3)2 – 625 c. x½– 9x¼ + 16 (x¼)2

  23. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625 (4x3)2 – 625 c. x½– 9x¼ + 16 (x¼)2– 9(x¼)

  24. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625 (4x3)2 – 625 c. x½– 9x¼ + 16 (x¼)2– 9(x¼) + 16

  25. Ex. 1 Write each expression in quadratic form, if possible. a. x4 + 13x2 + 36 (x2)2 + 13(x2) + 36 b. 16x6 – 625 (4x3)2 – 625 c. x½– 9x¼ + 16 (x¼)2– 9(x¼) + 16

  26. Ex. 2 Solve each equation. a. x4 = 16

  27. Ex. 2 Solve each equation. a. x4 = 16 -16 -16

  28. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0

  29. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

  30. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0

  31. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0

  32. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 )(x2 ) = 0

  33. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0

  34. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0

  35. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0

  36. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0

  37. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 (x )(x ) = 0

  38. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

  39. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0

  40. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

  41. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

  42. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

  43. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

  44. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 ORx2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

  45. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 ORx2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 x = 2 OR x = -2

  46. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 ORx2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4 x = 2 OR x = -2

  47. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 ORx2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4 x = 2 OR x = -2 OR x = ±2i

  48. Ex. 2 Solve each equation. a. x4 = 16 -16 -16 x4 – 16 = 0 (x2)2 – 16 = 0 ( )( ) = 0 (x2 – 4)(x2 + 4) = 0 x2 – 4 = 0 ORx2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 ORx2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x2 = -4 +2 +2 -2 -2 √x2 = √-4 x = 2 OR x = -2 OR x = ±2i

  49. b. x4 + 11x2 + 18 = 0

  50. b. x4 + 11x2 + 18 = 0

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