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Angstrom Care 1www.AngstromCare.com Angstrom Care Quadratic Equation II.

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Presentation on theme: "Angstrom Care 1www.AngstromCare.com Angstrom Care Quadratic Equation II."— Presentation transcript:

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2 Angstrom Care 1www.AngstromCare.com Angstrom Care Quadratic Equation II

3 Angstrom Care 2www.AngstromCare.com Quadratic Equations, Quadratic Functions and Absolute Values

4 Angstrom Care 3www.AngstromCare.com Solving a Quadratic Equation by factorization by graphical method by taking square roots by quadratic equation by using completing square

5 Angstrom Care 4www.AngstromCare.com By factorization roots (solutions)

6 Angstrom Care 5www.AngstromCare.com By graphical method x y O roots

7 Angstrom Care 6www.AngstromCare.com By taking square roots A quadratic equation must contain two roots. ?

8 Angstrom Care 7www.AngstromCare.com By taking square roots

9 Angstrom Care 8www.AngstromCare.com Solving a Quadratic Equation by the quadratic Formula

10 Angstrom Care 9www.AngstromCare.com By quadratic equation

11 Angstrom Care 10www.AngstromCare.com a =b =c =110-7

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

13 Angstrom Care 12www.AngstromCare.com Two distinct (unequal) real roots x-intercepts

14 Angstrom Care 13www.AngstromCare.com One double (repeated) real roots x-intercept

15 Angstrom Care 14www.AngstromCare.com No real roots no x-intercept

16 Angstrom Care 15www.AngstromCare.com Nature of Roots

17 Angstrom Care 16www.AngstromCare.com = b 2 - 4ac Since the expression b 2 - 4ac can be used to determine the nature of the roots of a quadratic equation in the form ax 2 – bx + c = 0, it is called the discriminant of the quadratic equation.

18 Angstrom Care 17www.AngstromCare.com Two distinct (unequal) real roots x-intercepts = b 2 - 4ac > 0

19 Angstrom Care 18www.AngstromCare.com One double (repeated) real roots x-intercept = b 2 - 4ac = 0

20 Angstrom Care 19www.AngstromCare.com No real roots no x-intercept = b 2 - 4ac < 0

21 Angstrom Care 20www.AngstromCare.com Solving a Quadratic Equation by Completing the Square

22 Angstrom Care 21www.AngstromCare.com Solving a Quadratic Equation by Completing the Square

23 Angstrom Care 22www.AngstromCare.com Relations between the Roots and the Coefficients

24 Angstrom Care 23www.AngstromCare.com If α and β(p and q, x 1 and x 2 ) are the roots of ax 2 + bx + c = 0, then sum of roots = α + β and product of roots = αβ

25 Angstrom Care 24www.AngstromCare.com Forming Quadratic Equations with Given Roots

26 Angstrom Care 25www.AngstromCare.com x 2 – (sum of the roots)x + (product of roots) = 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 x 2 + x – 6 = 0

27 Angstrom Care 26www.AngstromCare.com Linear Functions and Their Graphs

28 Angstrom Care 27www.AngstromCare.com

29 Angstrom Care 28www.AngstromCare.com c 0 x y O

30 Angstrom Care 29www.AngstromCare.com c 0 x y O

31 Angstrom Care 30www.AngstromCare.com

32 Angstrom Care 31www.AngstromCare.com

33 Angstrom Care 32www.AngstromCare.com c 0 x y O m 0 c

34 Angstrom Care 33www.AngstromCare.com c 0 x y O m 0 c

35 Angstrom Care 34www.AngstromCare.com c 0 x y O m 0 c

36 Angstrom Care 35www.AngstromCare.com c 0 x y O m 0 c

37 Angstrom Care 36www.AngstromCare.com c 0 x y O m 0 c

38 Angstrom Care 37www.AngstromCare.com

39 Angstrom Care 38www.AngstromCare.com Open upwards Vertex Open upwards Line of symmetry (a 0)

40 Angstrom Care 39www.AngstromCare.com Open downwards Vertex Line of symmetry (a 0)

41 Angstrom Care 40www.AngstromCare.com Vertex (Turning point) Local (Relative) Maximum point (max. pt.) Local (Relative) Minimum point point (mini. pt.)

42 Angstrom Care 41www.AngstromCare.com y = ax 2

43 Angstrom Care 42www.AngstromCare.com x y O y = ax 2 (a 0)

44 Angstrom Care 43www.AngstromCare.com x y O y = ax 2 + bx + c (a 0) b 2 - 4ac 0 roots 2 real roots (c 0)

45 Angstrom Care 44www.AngstromCare.com x y O (a 0) b 2 - 4ac 0 repeated roots root y = ax 2 + bx + c (c 0)

46 Angstrom Care 45www.AngstromCare.com x y O (a 0) B 2 - 4ac 0 No real roots y = ax 2 + bx + c (c 0) No intercept

47 Angstrom Care 46www.AngstromCare.com

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

49 Angstrom Care 48www.AngstromCare.com Absolute Values

50 Angstrom Care 49www.AngstromCare.com 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

51 Angstrom Care 50www.AngstromCare.com For all real numbers x and y, (y 0)

52 Angstrom Care 51www.AngstromCare.com Generalization If | x | = a, where a 0, then x = a or x = -a

53 Angstrom Care 52www.AngstromCare.com Thank you


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