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Chapter 3

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Chapter 3

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  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.”