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CHAPTER 7

CHAPTER 7. QUADRATIC EQUATIONS AND FUNCTIONS. 7-1. Completing the Square. Completing the Square. Transform the equation so that the constant term c is alone on the right side. If a , the coefficient of the second-degree term, is not equal to 1, then divide both sides by a .

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CHAPTER 7

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  1. CHAPTER 7 QUADRATIC EQUATIONS AND FUNCTIONS

  2. 7-1 Completing the Square

  3. Completing the Square • Transform the equation so that the constant term cis alone on the right side. • If a, the coefficient of the second-degree term, is not equal to 1, then divide both sides by a. • Add the square of half the coefficient of the first-degree term, (b/2a)2, to both sides (Completing the square)

  4. Completing the Square • Factor the left side as the square of a binomial. • Complete the solution using the fact that (x + q)2 = r is equivalent to • x + q = ±r

  5. Solve: a2 – 5a + 3= 0 • Move the 3 to the other side • a = 1 • Complete the square, add (5/2)2 • Factor • Solve

  6. Solve: x2 – 6x - 3= 0 • Move the 3 to the other side • a = 1 • Complete the square, add (3)2 • Factor • Solve

  7. Solve: 2y2 + 2y + 5 = 0 • move the 5 to the other side • divide both sides by 2 (a  1) • Add (1/2)2 to both sides • Factor • Solve

  8. Solve: 7x2 – 8x + 3 = 0 • move the 3 to the other side • divide both sides by 7 (a  1) • Add (4/7)2 to both sides • Factor • Solve

  9. 7-2 Quadratic Formula

  10. The Quadratic Formula The solutions of the quadratic equation ax2 + bx + c = 0 (a  0) are given by the formula:

  11. Solve • 3x2 + x – 1 = 0 • 5y2 = 6y – 3 • 2x2 – 3x + 7 = 0

  12. 7-3 The Discriminant

  13. Discriminant The discriminant is used to determine the nature of the roots of a quadratic equation and is equal to: D = b2 – 4ac

  14. Discriminant Cases • If D is positive, then the roots are real and unequal. • If D is zero, then the roots are real and equal (double root) . • negative, the roots are imaginary.

  15. Find the Discriminant • x2 + 6x – 2 = 0 • 3x2 – 4x√3 + 4 = 0 • x2 – 6x + 10 = 0

  16. Discriminant The discriminant also show you whether a quadratic equation with integral coefficients has rational roots. D = b2 – 4ac

  17. Test for Rational Roots • If a quadratic equation has integral coefficients and its discriminant is a perfect square, then the equation has rational roots.

  18. Test for Rational Roots • If the quadratic equation can be transformed into an equivalent equation that meets this test, then it has rational roots.

  19. Find the Determinant and Identify the Nature of the Roots • 3x2 - 7x + 5 = 0 • 2x2 - 13x + 15 = 0 • x2 + 6x + 10 = 0

  20. 7-4 Equations in Quadratic Form

  21. Quadratic Form An equation in quadratic form can be written as: a[f(x)]2 + b[(f(x)] + c = 0 where a 0 and f(x) is some function of x. It is helpful to replace f(x) with a single variable.

  22. Example (3x – 2)2 – 5(3x -2) – 6 = 0 Let z = 3x – 2, then z2 – 5z – 6 = 0 Solve for z and then solve for x

  23. Solve Using Quadratic Form • (x + 2)2 – 5(x + 2) – 14 = 0 • (3x + 4)2 + 6(3x + 4) – 16 = 0 • x4 + 7x2 – 18 = 0

  24. 7-5 Graphing y – k = a(x- h)2

  25. Parabola Parabola is the set of all points in the plane equidistant from a given line and a given point not on the line. Parabolas have an axis of symmetry (mirror image) either the x-axis or the y-axis. and

  26. Parabola The point where the parabola crosses it axis is the vertex.The graph is a smooth curve.

  27. Parabola The graph of an equation having the form y – k = a(x - h)2 has a vertex at (h, k) and its axis is the line x = h.

  28. Graph • y = x2 Use the form y – k = a(x – h)2 k= 0, h = 0, so the vertex is (0,0) and x = 0

  29. Table of Values

  30. Graph • y = ½x2 Use the form y – k = a(x – h)2 k= 0, h = 0, so the vertex is (0,0) and x = 0

  31. Graph • y = -½x2 Use the form y – k = a(x – h)2 k= 0, h = 0, so the vertex is (0,0) and x = 0

  32. Graph • The graph of y = ax2 opens upward if a> 0 and downward if a< 0. The larger the absolute value of a is, the “narrower” the graph.

  33. Graph • y = ½(x-3)2 Use the form y – k = a(x – h)2 k= 0, h = 3, so the vertex is (3,0) and x = 3

  34. Graph • To graph y = a(x – h)2, slide the graph of y = ax2 horizontally h units. If h > 0, slide it to the right; if h < 0, slide it to the left. The graph has vertex (h, 0) and its axis is the line x = h.

  35. Graph • y – 3 = ½x2 Use the form y – k = a(x – h)2 k= 3, h = 0, so the vertex is (0,3) and x = 0

  36. Graph • y + 3 = ½x2 Use the form y – k = a(x – h)2 k= -3, h = 0, so the vertex is (0,-3) and x = 0

  37. Graph • To graph y – k = ax2, slide the graph of y = ax2 vertically k units. If k > 0, slide it upward; if k < 0, slide it downward. The graph has vertex (0, k) and its axis is the line x = 0.

  38. 7-6 Quadratic Functions

  39. Quadratic Functions A function that can be written in either of two forms. General form: f(x) = ax2 + bx + c Completed square form: a(x-h)2 + k

  40. Graph • f(x) = 2(x – 3)2 + 1 y = 2(x – 3)2 + 1

  41. Graph • It’s a parabola with vertex (3,1) and axis x = 3

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