Complex Numbers

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
  readers consciousness as much as if it were in
  type on the page.

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Mathematics 116

• Complex Numbers

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Imaginary unit i
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Set of Complex Numbers

• R real numbers
• I imaginary numbers
• C Complex numbers

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Elbert Hubbard

• Positive anything is better than negative
  nothing.

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Standard Form of Complex number

• a bi
• Where a and b are real numbers
• 0 bi bi is a pure imaginary number

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Equality of Complex numbers

• abi c di
• iff
• a c and b d

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Powers of i
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Add and subtract complex s

• Add or subtract the real and imaginary parts of
  the numbers separately.

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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.

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Multiply Complex s

• Multiply as if two polynomials and combine like
  terms as in the FOIL
• Note i squared -1

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Complex Conjugates

• a bi is the conjugate of a bi
• The product is a rational number

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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.

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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.

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Mathematics 116

• Solving Quadratic Equations
• Algebraically
• This section contains much information

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Def Quadratic Function

• General Form
• a,b,c,are real numbers and a not equal 0

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Objective Solve quadratic equations

• Two distinct solutions
• One Solution double root
• Two complex solutions
• Solve for exact and decimal approximations

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Solving Quadratic Equation 1

• Factoring
• Use zero Factor Theorem
• Set to 0 and factor
• Set each factor equal to zero
• Solve
• Check

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Solving Quadratic Equation 2

• Graphing
• Solve for y
• Graph and look for x intercepts
• Can not give exact answers
• Can not do complex roots.

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Solving Quadratic Equations 3Square Root
Property

• For any real number c

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Sample problem
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Sample problem 2
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Solve quadratics in the form
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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

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Sample problem 3
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Sample problem 4
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Dorothy Broude

• Act as if it were impossible to fail.

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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.

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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

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Sample Problem
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Sample Problem complete the square 2
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Sample problem complete the square 3
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Objective

• Solve quadratic equations using the technique of
  completing the square.

36
Mary Kay Ash

• Aerodynamically, the bumble bee shouldnt be
  able to fly, but the bumble bee doesnt know it
  so it goes flying anyway.

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College AlgebraVery Important Concept!!!

• The
• Quadratic
• Formula

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Objective of A students

• Derive
• the
• Quadratic Formula.

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Quadratic Formula

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

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Sample problem quadratic formula 1
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Sample problem quadratic formula 2
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Sample problem quadratic formula 3
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Pearl S. Buck

• All things are possible until they are proved
  impossible and even the impossible may only be
  so, as of now.

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Methods for solving quadratic equations.

• 1. Factoring
• 2. Square Root Principle
• 3. Completing the Square
• 4. Quadratic Formula

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Discriminant

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

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Joseph De Maistre (1753-1821 French Philosopher

• It is one of mans curious idiosyncrasies to
  create difficulties for the pleasure of resolving
  them.

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Sum of Roots
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Product of Roots
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CalculatorPrograms

• ALGEBRA?QUADRATIC
• QUADB
• ALG2
• QUADRATIC

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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
  Im going to complete every pass.

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Objective

• Solve by Extracting Square Roots

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Objective Know and Prove the Quadratic Formula

• If a,b,c are real numbers and not equal to 0

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Objective Solve quadratic equations

• Two distinct solutions
• One Solution double root
• Two complex solutions
• Solve for exact and decimal approximations

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Objective Solve Quadratic Equations using
Calculator

• Graphically
• Numerically
• Programs
• ALGEBRAA
• QUADB
• ALG2
• others

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Objective Use quadratic equations to model
and solve applied, real-life problems.
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DAlembert French Mathematician

• The difficulties you meet will resolve
  themselves as you advance. Proceed, and light
  will dawn, and shine with increasing clearness on
  your path.

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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)

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Axis of symmetry

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

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Objective

• Graph, determine domain, range, y intercept, x
  intercept

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Parabola with vertex (h,k)

• Standard Form

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Standard Form of a Quadratic Function

• Graph is a parabola
• Axis is the vertical line x h
• Vertex is (h,k)
• agt0 graph opens upward
• alt0 graph opens downward

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Find Vertex

• x coordinate is
• y coordinate is

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Vertex of quadratic function
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Objective Find minimum and maximum values of
functions in real life applications.

• 1. Graphically
• 2. Algebraically
• Standard form
• Use vertex
• 3. Numerically

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Roger Maris, New York Yankees Outfielder

• You hit home runs not by chance but by
  preparation.

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Objective

• Solve Rational Equations
• Check for extraneous roots
• Graphically and algebraically

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Objective

• Solve equations involving radicals
• Solve Radical Equations
• Check for extraneous roots
• Graphically and algebraically

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Problem radical equation
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Problem radical equation
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Problem radical equation
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Objective

• Solve Equations
• Quadratic in Form

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Objective

• Solve equations
• involving
• Absolute Value

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ProcedureAbsolute Value equations

• 1.Isolate the absolute value
• 2. Set up two equations joined by orand so note
• 3. Solve both equations
• 4.Check solutions

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Elbert Hubbard

• Positive anything is better than negative
  nothing.

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Elbert Hubbard

• Positive anything is better than negative
  nothing.

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Addition Property of Inequality

• Addition of a constant
• If a lt b then a c lt b c

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Multiplication property of inequality

• If a lt b and c gt 0, then ac gt bc
• If a lt b and c lt 0, then ac gt bc

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Objective

• Solve Inequalities Involving Absolute Value.
• Remember lt uses AND
• Remember gt uses OR
• and/or need to be noted

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Objective Estimate solutions of inequalities
graphically.

• Two Ways
• Change inequality to and set to 0
• Graph in 2-space
• Or Use Test and Y with appropriate window

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Objective

• Solve Polynomial Inequalities
• Graphically
• Algebraically
• (graphical is better the larger the degree)

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Objectives

• Solve Rational Inequalities
• Graphically
• algebraically
• Solve models with inequalities

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Zig Ziglar

• Positive thinking wont let you do anything but
  it will let you do everything better than
  negative thinking will.

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Zig Ziglar

• Positive thinking wont let you do anything but
  it will let you do everything better than
  negative thinking will.

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Mathematics 116 RegressionContinued

• Explore data Quadratic Models and Scatter Plots

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Objectives

• Construct Scatter Plots
• By hand
• With Calculator
• Interpret correlation
• Positive
• Negative
• No discernible correlation

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Objectives

• Use the calculator to determine quadratic models
  for data.
• Graph quadratic model and scatter plot
• Make predictions based on model

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Napoleon Hill

• There are no limitations to the mind except
  those we acknowledge.

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