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Intermediate Algebra Chapter 11

Intermediate Algebra Chapter 11. Quadratic Equations. Willa Cather –U.S. novelist.

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Intermediate Algebra Chapter 11

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  1. Intermediate AlgebraChapter 11 • Quadratic Equations

  2. 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.

  3. Intermediate Algebra 11.1 • Special Methods

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

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

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

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

  8. Sample problem

  9. Sample problem 2

  10. Solve quadratics in the form

  11. 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

  12. Sample problem 3

  13. Sample problem 4

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

  15. Intermediate Algebra 11.1 Gay • Completing • the • Square

  16. 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.

  17. 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

  18. Sample Problem

  19. Sample Problem complete the square 2

  20. Sample problem complete the square #3

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

  22. 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.”

  23. Intermediate Algebra 11.2 • The • Quadratic • Formula

  24. Objective of “A” students • Derive • the • Quadratic Formula.

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

  26. Sample problem quadratic formula #1

  27. Sample problem quadratic formula #2

  28. Sample problem quadratic formula #3

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

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

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

  32. Sum of Roots

  33. Product of Roots

  34. CalculatorPrograms • ALGEBRAQUADRATIC • QUADB • ALG2 • QUADRATIC

  35. 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.”

  36. Intermediate Algebra 11.4 • Quadratic Inequalities

  37. Sample Problem quadratic inequalities #1

  38. Sample Problem quadric inequalities #2

  39. Sample Problem quadratic inequalities #3

  40. Sample Problem quadratic inequalities #4

  41. Sample Problem quadratic inequalities #5

  42. Intermediate Algebra 11.5-11.6 • Quadratic Functions

  43. 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.”

  44. 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)

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

  46. Objective • Graph, determine domain, range, y intercept, x intercept

  47. Parabola with vertex (h,k) • Standard Form

  48. Find Vertex • x coordinate is • y coordinate is

  49. 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

  50. Sample Problems - graph

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