7.3 quadratic techniques

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7.3 quadratic techniques

  1. 1. 7.3 Quadratic Techniques to Solve Polynomial Equations
  2. 2. Objectives <ul><li>Solve third and fourth degree equations that contain quadratic factors, and </li></ul><ul><li>Solve other non-quadratic equations that can be written in quadratic form. </li></ul>
  3. 3. Intro <ul><li>Some equations are not quadratic but can be written in a form that resembles a quadratic equation. For example, the equation x 4 – 20x 2 + 64 = 0 can be written as (x 2 ) 2 – 20x 2 + 64 = 0. Equations that can be written this way are said to be equations in quadratic form. </li></ul>
  4. 4. Definition of Quadratic Form <ul><li>For any numbers a, b, and c except a = 0, an equation that may be written as </li></ul><ul><li>a[f(x)] 2 + b[f(x)] + c = 0, where f(x) is some expression in x, is in quadratic form. </li></ul>
  5. 5. <ul><li>Once an equation is written in quadratic form, it can be solved by the methods you have already learned to use for solving quadratic equations. </li></ul>
  6. 6. Ex. 1: Solve x 4 – 13x 2 + 36 = 0 The solutions or roots are -3, 3, -2, and 2.
  7. 7. The graph of x 4 – 13x 2 + 36 = 0 looks like: The graph of y = x 4 – 13x 2 + 36 crosses the x-axis 4 times. There will be 4 real solutions.
  8. 8. <ul><li>Recall that (a m ) n = a mn for any positive number a and any rational numbers n and m. This property of exponents that you learned in chapter 5 is often used when solving equations. </li></ul>
  9. 9. Ex. 2: Solve
  10. 10. Ex. 3: Solve
  11. 11. Ex. 4: Solve
  12. 12. Ex. 4: Solve There is no real number x such that is = -1. The only solution would be 64.
  13. 13. <ul><li>Some cubic equations can be solved using the quadratic formula. First a binomial factor must be found. </li></ul>
  14. 14. Ex. 5: Solve
  15. 15. Thank you Mrs. Spitz for the wonderful ppt.
  16. 16. Reminder <ul><li>You can always ask me questions before and after class </li></ul><ul><li>Chapter 5, 6, and 7 test – Friday </li></ul><ul><li>NO class on Monday July 5. </li></ul>

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