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

Imaginary unit i

2

1

1

i

i

Set of Complex Numbers

• R = real numbers

• I = imaginary numbers

• C = Complex numbers

R I C

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

Powers of i

1

2

3

4

1

1

i i

i

i i

i

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

2 1i

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

2( )f x ax bx c

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

2if x c then

x c or x c

x c

Sample problem

2 40x 40x

4 10x

2 10x

Sample problem 225 2 62x

25 60x 2 12x

12x 2 3x

Solve quadratics in the form

2ax b 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

Sample problem 3

23 16x

3 16x 3 4x

3 4 3 4 3 4x x or x

1 7 1, 7x or x

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

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

2x bx k

Sample Problem

2 8 5 0x x

4 11x

Sample Problem complete the square 2

2 5 1 0x x 5 29

2x

Sample problem complete the square #3

23 7 10 4x x

7 23

6 6x i

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

23 8 7 0

4 5

3 3

x x

x i

2 4

2

b b acx

a

Sample problem quadratic formula #1

22 9 5 0x x 1

, 52

Sample problem quadratic formula #2

2 12 4 0x x

6 2 10x

Sample problem quadratic formula #3

23 8 7 0x x 4 5

3 3x i

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

2 4b ac

Joseph De Maistre (1753-1821 – French Philosopher

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

Sum of Roots

1 2

br r

a

Product of Roots

1 2

cr r

a

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

2 0If a c where c

then a c

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

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–“The difficulties you meet will

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

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)

Axis of symmetry

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

• Equation is x = constant

Objective

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

2

2

y x

y ax

Parabola with vertex (h,k)

• Standard Form

2y a x h k

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

Find Vertex

• x coordinate is

• y coordinate is

2

b

a

2

bf

a

Vertex of quadratic function

,2 2

b bf

a a

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

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

Objective:

• Solve Rational Equations

–Check for extraneous roots

–Graphically and algebraically

Objective

• Solve equations involving radicals

–Solve Radical Equations

Check for extraneous roots

–Graphically and algebraically

Problem: radical equation

3 2 4 2 0x

6

Problem: radical equation

1 7x x

10

Problem: radical equation

2 3 2 2x x

23

Objective:

• Solve Equations

• Quadratic in Form

Objective

• Solve equations

• involving

• Absolute Value

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

Elbert Hubbard

• “Positive anything is better than negative nothing.”

Elbert Hubbard

• “Positive anything is better than negative nothing.”

Addition Property of Inequality

• Addition of a constant

• If a < b then a + c < b + c

Multiplication property of inequality

• If a < b and c > 0, then ac > bc

• If a < b and c < 0, then ac > bc

Objective:

• Solve Inequalities Involving Absolute Value.

• Remember < uses “AND”

• Remember > uses “OR”

• and/or need to be noted

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

Objective:

• Solve Polynomial Inequalities

–Graphically

–Algebraically

–(graphical is better the larger the degree)

Objectives:

• Solve Rational Inequalities

–Graphically

–algebraically

• Solve models with inequalities

Zig Ziglar

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

Zig Ziglar

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

Mathematics 116 RegressionContinued

• Explore data: Quadratic Models and Scatter Plots

Objectives• Construct Scatter Plots

– By hand

– With Calculator

• Interpret correlation

– Positive

– Negative

– No discernible correlation

Objectives:

• Use the calculator to determine quadratic models for data.

• Graph quadratic model and scatter plot

• Make predictions based on model

Napoleon Hill

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