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Chapter 3 • Complex Numbers • Quadratic Functions and Equations • Inequalities • Rational Equations • Radical Equations • Absolute Value Equations

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Page 1: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Chapter 3

• Complex Numbers

• Quadratic Functions and Equations

• Inequalities

• Rational Equations

• Radical Equations

• Absolute Value Equations

Page 2: 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.

Page 3: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Mathematics 116

•Complex Numbers

Page 4: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Imaginary unit i

2

1

1

i

i

Page 5: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Set of Complex Numbers

• R = real numbers

• I = imaginary numbers

• C = Complex numbers

R I C

Page 6: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Elbert Hubbard

–“Positive anything is better than negative nothing.”

Page 7: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Standard Form of Complex number

• a + bi

• Where a and b are real numbers

• 0 + bi = bi is a pure imaginary number

Page 8: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Equality of Complex numbers

• a+bi = c + di

• iff

• a = c and b = d

Page 9: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Powers of i

1

2

3

4

1

1

i i

i

i i

i

Page 10: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Add and subtract complex #s

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

Page 11: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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.”

Page 12: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Multiply Complex #s

• Multiply as if two polynomials and combine like terms as in the FOIL

• Note i squared = -1

2 1i

Page 13: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Complex Conjugates

• a – bi is the conjugate of a + bi

• The product is a rational number

Page 14: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Divide Complex #s

• Multiply numerator and denominator by complex conjugate of denominator.

• Write answer in standard form

Page 15: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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.”

Page 16: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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.

Page 17: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Mathematics 116

• Solving Quadratic Equations

• Algebraically• This section contains much

information

Page 18: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Def: Quadratic Function

• General Form

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

2( )f x ax bx c

Page 19: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective – Solve quadratic equations

• Two distinct solutions

• One Solution – double root

• Two complex solutions

• Solve for exact and decimal approximations

Page 20: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Solving Quadratic Equation #1

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

Page 21: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Solving Quadratic Equation #2

• Graphing

• Solve for y

• Graph and look for x intercepts

• Can not give exact answers

• Can not do complex roots.

Page 22: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Solving Quadratic Equations #3Square Root Property

• For any real number c

2if x c then

x c or x c

x c

Page 23: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem

2 40x 40x

4 10x

2 10x

Page 24: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem 225 2 62x

25 60x 2 12x

12x 2 3x

Page 25: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Solve quadratics in the form

2ax b c

Page 26: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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

Page 27: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem 3

23 16x

3 16x 3 4x

3 4 3 4 3 4x x or x

1 7 1, 7x or x

Page 28: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem 4

27 25 2 3 0x

225 2 3 7x 2 7

2 325

x

7 72 3

25 5x i

3 71.5 0.26

2 10x i i

Page 29: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Dorothy Broude

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

Page 30: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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.

Page 31: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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

2x bx k

Page 32: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample Problem

2 8 5 0x x

4 11x

Page 33: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample Problem complete the square 2

2 5 1 0x x 5 29

2x

Page 34: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem complete the square #3

23 7 10 4x x

7 23

6 6x i

Page 35: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective:

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

Page 36: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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.”

Page 37: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

College AlgebraVery Important Concept!!!

•The

•Quadratic

•Formula

Page 38: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective of “A” students

• Derive

• the

• Quadratic Formula.

Page 39: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Quadratic Formula

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

23 8 7 0

4 5

3 3

x x

x i

2 4

2

b b acx

a

Page 40: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem quadratic formula #1

22 9 5 0x x 1

, 52

Page 41: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem quadratic formula #2

2 12 4 0x x

6 2 10x

Page 42: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sample problem quadratic formula #3

23 8 7 0x x 4 5

3 3x i

Page 43: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Pearl S. Buck

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

Page 44: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Methods for solving quadratic equations.

• 1. Factoring

• 2. Square Root Principle

• 3. Completing the Square

• 4. Quadratic Formula

Page 45: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Discriminant

• Negative – complex conjugates• Zero – one rational solution (double

root)• Positive

– Perfect square – 2 rational solutions– Not perfect square – 2 irrational

solutions

2 4b ac

Page 46: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Joseph De Maistre (1753-1821 – French Philosopher

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

Page 47: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Sum of Roots

1 2

br r

a

Page 48: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Product of Roots

1 2

cr r

a

Page 49: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

CalculatorPrograms

• ALGEBRAQUADRATIC

• QUADB

• ALG2

• QUADRATIC

Page 50: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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.”

Page 51: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective

• Solve by Extracting Square Roots

2 0If a c where c

then a c

Page 52: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective: Know and Prove the Quadratic Formula

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

2 4

2

b b acx

a

Page 53: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective – Solve quadratic equations

• Two distinct solutions

• One Solution – double root

• Two complex solutions

• Solve for exact and decimal approximations

Page 54: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective: Solve Quadratic Equations using Calculator

• Graphically• Numerically• Programs

– ALGEBRAA– QUADB– ALG2– others

Page 55: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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

Page 56: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

D’Alembert – 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.”

Page 57: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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)

Page 58: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Axis of symmetry

• The vertical line that goes through the vertex of the parabola.

• Equation is x = constant

Page 59: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective

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

2

2

y x

y ax

Page 60: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Parabola with vertex (h,k)

• Standard Form

2y a x h k

Page 61: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Standard Form of a Quadratic Function

• 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

2( ) ( )f x a x h k

Page 62: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Find Vertex

• x coordinate is

• y coordinate is

2

b

a

2

bf

a

Page 63: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Vertex of quadratic function

,2 2

b bf

a a

Page 64: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective: Find minimum and maximum values of functions in real

life applications.

• 1. Graphically

• 2. Algebraically

–Standard form

–Use vertex

3. Numerically

Page 65: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Roger Maris, New York Yankees Outfielder

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

Page 66: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective:

• Solve Rational Equations

–Check for extraneous roots

–Graphically and algebraically

Page 67: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective

• Solve equations involving radicals

–Solve Radical Equations

Check for extraneous roots

–Graphically and algebraically

Page 68: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Problem: radical equation

3 2 4 2 0x

6

Page 69: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Problem: radical equation

1 7x x

10

Page 70: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Problem: radical equation

2 3 2 2x x

23

Page 71: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective:

• Solve Equations

• Quadratic in Form

Page 72: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective

• Solve equations

• involving

• Absolute Value

Page 73: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Procedure:Absolute Value equations

• 1.Isolate the absolute value• 2. Set up two equations joined

by “or”and so note• 3. Solve both equations• 4.Check solutions

Page 74: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Elbert Hubbard

• “Positive anything is better than negative nothing.”

Page 75: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Elbert Hubbard

• “Positive anything is better than negative nothing.”

Page 76: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Addition Property of Inequality

• Addition of a constant

• If a < b then a + c < b + c

Page 77: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Multiplication property of inequality

• If a < b and c > 0, then ac > bc

• If a < b and c < 0, then ac > bc

Page 78: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective:

• Solve Inequalities Involving Absolute Value.

• Remember < uses “AND”

• Remember > uses “OR”

• and/or need to be noted

Page 79: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

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

Page 80: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objective:

• Solve Polynomial Inequalities

–Graphically

–Algebraically

–(graphical is better the larger the degree)

Page 81: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objectives:

• Solve Rational Inequalities

–Graphically

–algebraically

• Solve models with inequalities

Page 82: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Zig Ziglar

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

Page 83: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Zig Ziglar

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

Page 84: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Mathematics 116 RegressionContinued

• Explore data: Quadratic Models and Scatter Plots

Page 85: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objectives• Construct Scatter Plots

– By hand

– With Calculator

• Interpret correlation

– Positive

– Negative

– No discernible correlation

Page 86: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Objectives:

• Use the calculator to determine quadratic models for data.

• Graph quadratic model and scatter plot

• Make predictions based on model

Page 87: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations

Napoleon Hill

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

Page 88: Chapter 3 Complex Numbers Quadratic Functions and Equations Inequalities Rational Equations Radical Equations Absolute Value Equations