1. Angstrom Care 培苗社 Quadratic Equation II
積極推崇『我要學』的心態，
糾正『要我學』的被動心態。

2. Quadratic Equations, Quadratic Functions and Absolute Values

3. Solving a Quadratic Equation
by factorization
by graphical method
by taking square roots
by quadratic equation
by using completing square

4. By factorization
roots (solutions)

5. By graphical method y
roots x O

6. ? By taking square roots A quadratic equation must contain two roots.

7. By taking square roots

8. Solving a Quadratic Equation by the quadratic Formula

9. By quadratic equation

10. a = 1
b = -7
c = 10

11. In general, a quadratic equation may have :
(1) two distinct (unequal) real roots
(2) one double (repeated) real root
(3) no real roots

12. Two distinct (unequal) real roots
x-intercepts

13. One double (repeated) real roots
x-intercept

14. No real roots
no x-intercept

15. Nature of Roots

16. △ = b2 - 4ac
Since the expression b2 - 4ac can be used to determine the nature of the roots of a quadratic equation in the form ax2 – bx + c = 0, it is called the discriminant of the quadratic equation.

17. Two distinct (unequal) real roots
△ = b2 - 4ac > 0
x-intercepts

18. One double (repeated) real roots
△ = b2 - 4ac = 0
x-intercept

19. No real roots
△ = b2 - 4ac < 0
no x-intercept

20. Solving a Quadratic Equation by Completing the Square

21. Solving a Quadratic Equation by Completing the Square

22. Relations between the Roots and the Coefficients

23. If α and β(p and q, x1 and x2) are the roots of ax2 + bx + c = 0,
then sum of roots =
α + β
and product of roots = αβ

24. Forming Quadratic Equations with Given Roots

25. x = 2 or x = -3 x – 2 = 0 or x + 3 = 0 (x – 2)(x + 3) = 0
Forming Quadratic Equations with Given Roots
In S.3, when α = 2 and β = -3
x = 2 or x = -3
x – 2 = 0 or x + 3 = 0
(x – 2)(x + 3) = 0
x2 + x – 6 = 0
x2 – (sum of the roots)x + (product of roots) = 0

26. Linear Functions and Their Graphs

28. y
c＞0 x O

29. y x O
c＜0

30. Linear Functions

32. y
m＞0
c＞0 c x O

33. y
m＞0
c＜0 x O c

34. y c x O
m＜0
c＞0

35. y x O c
m＜0
c＜0

36. y O x c
m＜0
c＝0

38. Open upwards Open upwards (a＞0) Vertex Line of symmetry

39. Vertex
Open downwards
Line of symmetry
(a＞0)

40. Vertex (Turning point)
Local (Relative) Maximum point (max. pt.)
Local (Relative) Minimum point point (mini. pt.)

41. y = ax2

42. y
y = ax2
(a＞0) x O

43. y y = ax2 + bx + c b2 - 4ac＞0 2 real roots (a＞0) (c＜0) x roots O

44. y y = ax2 + bx + c b2 - 4ac＝0 repeated roots (a＞0) (c＞0) x root O

45. y y = ax2 + bx + c B2 - 4ac＜0 No real roots (a＞0) (c＞0) x No intercept

47. mini value of the function = -1 mini point = (-2, -1)
Finding the turning point of a Quadratic Function by Completing the Square
Because a = +ve, there exists a minimum point.
mini value of the function = -1
mini point = (-2, -1)

48. Absolute Values

49. Let x be any real number. The absolute value of x, denoted by | x |, is defined as
x if x ≧ 0.
-x if x < 0.
eg. | 5 | = 5, | 0 | = 0, | -5 | = 5

50. For all real numbers x and y,

51. If | x | = a, where a ≧0, then x = a or x = -a
Generalization
If | x | = a, where a ≧0,
then x = a or x = -a

52. Thank you