1. QUADRATIC EQUATIONS AND FUNCTIONS
CHAPTER 7
QUADRATIC EQUATIONS AND FUNCTIONS

2. 7-1
Completing the Square

3. 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.
Add the square of half the coefficient of the first-degree term, (b/2a)2, to both sides (Completing the square)

4. Factor the left side as the square of a binomial.
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. Equations in Quadratic Form
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 1 -1 2 4 -2

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

33. 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

35. 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.

37. 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

40. 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.

41. 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

43. 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

45. 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.

46. 7-6
Quadratic Functions

47. Completed square form:
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

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

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

51. Graph
f(x) = 3x2 – 6x + 1
y = 3x2 – 6x + 1

52. Graph
y = 3x2 – 6x + 1
Rewrite the equation in the form : y – k = a(x – h)2
y -1 = 3(x2 – 2x)
y – = 3(x2 – 2x + 1)
y + 2 = 3(x-1)2

54. Quadratic Functions
Let f(x) = ax2 + bx + c, a0
If a < 0, f has a maximum value.
If a > 0, f has a minimum value and

55. Quadratic Functions
The graph of f is a parabola. This maximum or minimum value of f is the y-coordinate when x = - b/2a, at the vertex of the graph.

56. Quadratic Functions
Let f(x) = ax2 + bx + c, a0
This maximum or minimum value of f is the y-coordinate when
x = - b/2a, at the vertex of the graph.

57. Example
y = 1/2x2 + 3x – 7/4
Find the maximum or minimum value of f.
Find the vertex of the graph of f

58. Example
y = 1/2x2 + 3x – 7/4
a = ½, a > 0, then f has a minimum value.
Minimum occurs when
x = -b/2a
x = -3/2*1/2 = -3

59. Example
y = 1/2x2 + 3x – 7/4
y = ½(-3)2 + 3(-3) – 7/4
y = -25/4
So the minimum value of f is
-25/4 and the vertex is
(-3, -25/4)

60. Writing Quadratic Equations and Functions
7-7
Writing Quadratic Equations and Functions

61. and Theorem A quadratic equation with roots r1 and r2 is
x2 – (r1 + r2)x + r1r2 = 0 or
a[x2 – (r1 + r2)x + r1r2 ] = 0
and

62. Theorem
The equation just given is equivalent to
a[x2 – (sum of roots)x + product of roots ] = 0

63. Example Find a quadratic equation with roots (2 + i)/3 and (2 – i)/3
Sum of roots =
(2 + i)/3 + (2 – i)/3 = 4/3
Product of roots =
(2 + i)/3 (2 – i)/3= 5/9
x2 – 4/3x + 5/9 = 9x2 -12x + 5

64. Theorem If r1 and r2 are the roots of a quadratic equation
ax2 + bx + c =0, then
r1 + r2 = sum of roots = -b/a
and
r1r2 = product of roots = c/a

65. Example Find the roots of 2x2 + 9x + 5 = 0 r1 = -9 + 41 r2 = -9 - 41

66. Check 4 4 = -18/4 = - b/a r1 · r2 = -9 + 41 · -9 - 41 4 4
= -18/4 = - b/a
r1 · r2 = -9 + 41 · -9 - 41
= 5/2 = c/a

67. END