2. Intermediate Algebra Chapter 8 Quadratic Equations

3. Willa Cather –U.S. novelist “Art, it seems to me, should simplify. That indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the whole – so that all one has suppressed and cut away is there to the reader’s consciousness as much as if it were in type on the page.

4. Intermediate Algebra 8.1 Special Methods

5. Def: Quadratic Function General Form a,b,c,are real numbers and a not equal 0

6. Solving Quadratic Equation #1 Factoring Use zero Factor Theorem Set = to 0 and factor Set each factor equal to zero Solve Check

7. Solving Quadratic Equation #2 Graphing Solve for y Graph and look for x intercepts Can not give exact answers Can not do complex roots.

8. Solving Quadratic Equations #3 Square Root Property For any real number c

9. Sample problem

10. Sample problem 2

11. Solve quadratics in the form

12. Procedure 1. Use LCD and remove fractions 2. Isolate the squared term 3. Use the square root property 4. Determine two roots 5. Simplify if needed

13. Sample problem 3

14. Sample problem 4

15. Dorothy Broude “Act as if it were impossible to fail.”

16. Intermediate Algebra 8.1 Gay Completing the Square

17. Completing the square informal Make one side of the equation a perfect square and the other side a constant. Then solve by methods previously used.

18. Procedure: Completing the Square 1. If necessary, divide so leading coefficient of squared variable is 1. 2. Write equation in form 3. Complete the square by adding the square of half of the linear coefficient to both sides. 4. Use square root property 5. Simplify

19. Sample Problem

20. Sample Problem complete the square 2

21. Sample problem complete the square #3

22. Objective: Solve quadratic equations using the technique of completing the square.

23. Mary Kay Ash “Aerodynamically, the bumble bee shouldn’t be able to fly, but the bumble bee doesn’t know it so it goes flying anyway.”

24. Intermediate Algebra 8.2 The Quadratic Formula

25. Objective of “A” students Derive the Quadratic Formula.

26. Quadratic Formula For all a,b, and c that are real numbers and a is not equal to zero

27. Sample problem quadratic formula #1

28. Sample problem quadratic formula #2

29. Sample problem quadratic formula #3

30. Pearl S. Buck “All things are possible until they are proved impossible and even the impossible may only be so, as of now.”

31. Methods for solving quadratic equations. 1. Factoring 2. Square Root Principle 3. Completing the Square 4. Quadratic Formula

32. Discriminant Negative – complex conjugates Zero – one rational solution (double root) Positive –Perfect square – 2 rational solutions –Not perfect square – 2 irrational solutions

33. Sum of Roots

34. Product of Roots

35. Calculator Programs ALGEBRA  QUADRATIC QUADB ALG2 QUADRATIC

36. Harry Truman – American President “A pessimist is one who makes difficulties of his opportunities and an optimist is one who makes opportunities of his difficulties.”

37. Intermediate Algebra 8.4 Quadratic Inequalities

38. Sample Problem quadratic inequalities #1

39. Sample Problem quadric inequalities #2

40. Sample Problem quadratic inequalities #3

41. Sample Problem quadratic inequalities #4

42. Sample Problem quadratic inequalities #5

43. Intermediate Algebra 8.5-8.6 Quadratic Functions

44. Orison Swett Marden “All who have accomplished great things have had a great aim, have fixed their gaze on a goal which was high, one which sometimes seemed impossible.”

45. Vertex The point on a parabola that represents the absolute minimum or absolute maximum – otherwise known as the turning point. y coordinate determines the range. (x,y)

46. Axis of symmetry The vertical line that goes through the vertex of the parabola. Equation is x = constant

47. Objective Graph, determine domain, range, y intercept, x intercept

48. Parabola with vertex (h,k) Standard Form

49. Find Vertex x coordinate is y coordinate is

50. Graphing Quadratic 1. Determine if opens up or down 2. Determine vertex 3. Determine equation of axis of symmetry 4. Determine y intercept 5. Determine point symmetric to y intercept 6. Determine x intercepts 7. Graph

51. Sample Problems - graph

53. Roger Maris, New York Yankees Outfielder “You hit home runs not by chance but by preparation.”