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Chapter 3
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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
i  1
i  1
2
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
i i
1
i  1
2
i  i
3
i 1
4
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
i  1
2
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
f ( x)  ax  bx  c
2
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
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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 #3
Square Root Property
• For any real number c
if
x  c
2
x c
or
then
x c
x c
Sample problem
x  40
2
x   40
x   4  10
x  2 10
Sample problem 2
5x  2  62
2
5x  60
2
x  12
2
x   12
x  2 3
Solve quadratics in the form
 ax  b 
2
c
Procedure
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•
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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
 x  3
2
 16
x  3   16
x  3  4
x  3  4  x  3  4 or x  3  4
x  1 or x  7
1, 7
Sample problem 4
7  25  2 x  3  0
2
25  2 x  3  7
2
 2 x  3
7
7
2x  3  

i
25
5
2
7

25
3
7
x

i  1.5  0.26i
2
10
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 x 2  bx  k
• 3. Complete the square by adding the
square of half of the linear coefficient to
both sides.
• 4. Use square root property
• 5. Simplify
Sample Problem
x  8x  5  0
2
x  4  11
Sample Problem complete the square 2
x  5x  1  0
2
5  29
x
2
Sample problem complete the square #3
3x  7 x  10  4
2
7
23
x

i
6
6
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 Algebra
Very Important Concept!!!
•The
• Quadratic
•Formula
Objective of “A” students
• Derive
• the
• Quadratic Formula.

4
5

i
3
3
Quadratic Formula
• For all a,b, and c that are real numbers and a
is not equal to zero
b  b  4ac
x
2a
2
Sample problem quadratic formula #1
2x  9x  5  0
2
1

 , 5
2

Sample problem quadratic formula #2
x  12 x  4  0
2
x  6  2 10
Sample problem quadratic formula #3
3x  8 x  7  0
2
4
5
x

i
3
3
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.
• 2.
• 3.
• 4.
Factoring
Square Root Principle
Completing the Square
Quadratic Formula
Discriminant
b  4ac
2
• 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.”
Sum of Roots
b
r1  r2 
a
Product of Roots
c
r1 r2 
a
Calculator
Programs
• ALGEBRAQUADRATIC
• 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
If a  c where c  0
2
then
a c
Objective: Know and Prove the
Quadratic Formula
If a,b,c are real numbers and not equal to 0
b  b  4ac
x
2a
2
Objective – Solve quadratic
equations
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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
yx
2
y  ax
2
Parabola with vertex (h,k)
• Standard Form
y  a  x  h  k
2
Standard Form of a Quadratic
Function
f ( x)  a( x  h)  k
2
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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
• x coordinate is
• y coordinate is
b
2a
 b 
f 
 2a 
Vertex of quadratic function
 b
,
f
 2a

 b  
 
 2a  
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
2x  4  2  0
6
Problem: radical equation
x 1  x  7
10
Problem: radical equation
2x  3  x  2  2
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 Regression
Continued
• 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.”