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Chapter 3

- Complex Numbers
- Quadratic Functions and Equations
- Inequalities
- Rational Equations
- Radical Equations
- Absolute Value Equations

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.

Mathematics 116

- Complex Numbers

Set of Complex Numbers

- R = real numbers
- I = imaginary numbers
- C = Complex numbers

Elbert Hubbard

- “Positive anything is better than negative nothing.”

Standard Form of Complex number

- a + bi
- Where a and b are real numbers
- 0 + bi = bi is a pure imaginary number

Equality of Complex numbers

- a+bi = c + di
- iff
- a = c and b = d

Add and subtract complex #s

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

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.”

Multiply Complex #s

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

Complex Conjugates

- a – bi is the conjugate of a + bi
- The product is a rational number

Divide Complex #s

- Multiply numerator and denominator by complex conjugate of denominator.
- Write answer in standard form

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.”

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.

Mathematics 116

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

Def: Quadratic Function

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

Objective – Solve quadratic equations

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

Solving Quadratic Equation #1

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

Solving Quadratic Equation #2

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

Solving Quadratic Equations #3Square Root Property

- For any real number c

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

Dorothy Broude

- “Act as if it were impossible to fail.”

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.

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

Objective:

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

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.”

College AlgebraVery Important Concept!!!

- The
- Quadratic
- Formula

Objective of “A” students

- Derive
- the
- Quadratic Formula.

Quadratic Formula

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

Pearl S. Buck

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

Methods for solving quadratic equations.

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

Discriminant

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

Joseph De Maistre (1753-1821 – French Philosopher

- “It is one of man’s curious idiosyncrasies to create difficulties for the pleasure of resolving them.”

CalculatorPrograms

- ALGEBRAQUADRATIC
- QUADB
- ALG2
- QUADRATIC

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.”

Objective

- Solve by Extracting Square Roots

Objective: Know and Prove the Quadratic Formula

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

Objective – Solve quadratic equations

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

Objective: Solve Quadratic Equations using Calculator

- Graphically
- Numerically
- Programs
- ALGEBRAA
- QUADB
- ALG2
- others

Objective: Use quadratic equations to model and solve applied, real-life problems.

D’Alembert – French Mathematician applied, real-life problems.

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

Vertex applied, real-life problems.

- 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)

Axis of symmetry applied, real-life problems.

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

Objective applied, real-life problems.

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

Parabola with vertex (h,k) applied, real-life problems.

- Standard Form

Standard Form of a Quadratic Function applied, real-life problems.

- Graph is a parabola
- Axis is the vertical line x = h
- Vertex is (h,k)
- a>0 graph opens upward
- a<0 graph opens downward

Find Vertex applied, real-life problems.

- x coordinate is
- y coordinate is

Vertex of quadratic function applied, real-life problems.

Objective: Find minimum and maximum values of functions in real life applications.

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

Roger Maris, New York Yankees Outfielder real life applications.

- “You hit home runs not by chance but by preparation.”

Objective: real life applications.

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

Objective real life applications.

- Solve equations involving radicals
- Solve Radical Equations
Check for extraneous roots

- Graphically and algebraically

- Solve Radical Equations

Problem: radical equation real life applications.

Problem: radical equation real life applications.

Problem: radical equation real life applications.

Objective: real life applications.

- Solve Equations
- Quadratic in Form

Objective real life applications.

- Solve equations
- involving
- Absolute Value

Procedure:Absolute Value equations real life applications.

- 1.Isolate the absolute value
- 2. Set up two equations joined by “or”and so note
- 3. Solve both equations
- 4.Check solutions

Elbert Hubbard real life applications.

- “Positive anything is better than negative nothing.”

Elbert Hubbard real life applications.

- “Positive anything is better than negative nothing.”

Addition Property of Inequality real life applications.

- Addition of a constant
- If a < b then a + c < b + c

Multiplication property of inequality real life applications.

- If a < b and c > 0, then ac > bc
- If a < b and c < 0, then ac > bc

Objective: real life applications.

- Solve Inequalities Involving Absolute Value.
- Remember < uses “AND”
- Remember > uses “OR”
- and/or need to be noted

Objective: Estimate solutions of inequalities graphically. real life applications.

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

Objective: real life applications.

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

Objectives: real life applications.

- Solve Rational Inequalities
- Graphically
- algebraically

- Solve models with inequalities

Zig Ziglar real life applications.

- “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”

Zig Ziglar real life applications.

- “Positive thinking won’t let you do anything but it will let you do everything better than negative thinking will.”

Mathematics 116 Regression real life applications.Continued

- Explore data: Quadratic Models and Scatter Plots

Objectives real life applications.

- Construct Scatter Plots
- By hand
- With Calculator

- Interpret correlation
- Positive
- Negative
- No discernible correlation

Objectives: real life applications.

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

Napoleon Hill real life applications.

- “There are no limitations to the mind except those we acknowledge.”

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