1 / 88

Chapter 3

Chapter 3. Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations. Willa Cather –U.S. novelist.

Thomas
Télécharger la présentation

Chapter 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Chapter 3 • Complex Numbers • Quadratic Functions and Equations • Inequalities • Rational Equations • Radical Equations • Absolute Value 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. Mathematics 116 • Complex Numbers

  4. Imaginary unit i

  5. Set of Complex Numbers • R = real numbers • I = imaginary numbers • C = Complex numbers

  6. Elbert Hubbard • “Positive anything is better than negative nothing.”

  7. Standard Form of Complex number • a + bi • Where a and b are real numbers • 0 + bi = bi is a pure imaginary number

  8. Equality of Complex numbers • a+bi = c + di • iff • a = c and b = d

  9. Powers of i

  10. Add and subtract complex #s • Add or subtract the real and imaginary parts of the numbers separately.

  11. 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.”

  12. Multiply Complex #s • Multiply as if two polynomials and combine like terms as in the FOIL • Note i squared = -1

  13. Complex Conjugates • a – bi is the conjugate of a + bi • The product is a rational number

  14. Divide Complex #s • Multiply numerator and denominator by complex conjugate of denominator. • Write answer in standard form

  15. 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.”

  16. Calculator and Complex #s • Use Mode – Complex • Use i second function of decimal point • Use [Math][Frac] and place in standard form a + bi • Can add, subtract, multiply, and divide complex numbers with calculator.

  17. Mathematics 116 • Solving Quadratic Equations • Algebraically • This section contains much information

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

  19. Objective – Solve quadratic equations • Two distinct solutions • One Solution – double root • Two complex solutions • Solve for exact and decimal approximations

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

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

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

  23. Sample problem

  24. Sample problem 2

  25. Solve quadratics in the form

  26. 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

  27. Sample problem 3

  28. Sample problem 4

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

  30. 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.

  31. 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

  32. Sample Problem

  33. Sample Problem complete the square 2

  34. Sample problem complete the square #3

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

  36. 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.”

  37. College AlgebraVery Important Concept!!! • The • Quadratic • Formula

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

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

  40. Sample problem quadratic formula #1

  41. Sample problem quadratic formula #2

  42. Sample problem quadratic formula #3

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

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

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

  46. Joseph De Maistre (1753-1821 – French Philosopher • “It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.”

  47. Sum of Roots

  48. Product of Roots

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

  50. Ron Jaworski • “Positive thinking is the key to success in business, education, pro football, anything that you can mention. I go out there thinking that I’m going to complete every pass.”

More Related