Intermediate Algebra
Chapter 8
•Quadratic 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.
Intermediate Algebra 8.1
•Special Methods
Def: Quadratic Function
• General Form
• a,b,c,are real numbers and a not equal 0
f ( x)  ax  bx  c
2
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 #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
•
•
•
•
•
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.”
Intermediate Algebra 8.1 Gay
•Completing
•the
•Square
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.”
Intermediate Algebra 8.2
•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
Sum of Roots
b
r1  r2 
a
Product of Roots
c
r1 r2 
a
Calculator
Programs
• ALGEBRAQUADRATIC
• QUADB
• ALG2
• QUADRATIC
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.”
Intermediate Algebra 8.4
•Quadratic
Inequalities
Sample Problem quadratic
inequalities #1
x  2x  8  0
2
 2,4
Sample Problem quadric
inequalities #2
6x  x  2
2
1


,


2

2 
,

 3 
Sample Problem quadratic
inequalities #3
x  6x  9  0
R   ,  
2
Sample Problem quadratic
inequalities #4
x4
0
x 1
(1,4]
Sample Problem quadratic
inequalities #5
3
2

0
x2 x3
 2,3  5, 
Intermediate Algebra 8.5-8.6
•Quadratic Functions
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.”
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
Find Vertex
• x coordinate is
• y coordinate is
b
2a
 b 
f 
 2a 
Graphing Quadratic
• 1. Determine if opens up or down
• 2. Determine vertex
• 3. Determine equation of axis of
symmetry
• 4. Determine y intercept
• 5. Determine point symmetric to y
intercept
• 6. Determine x intercepts
• 7. Graph
Sample Problems - graph
y  x  6x  5
2
y   x  2x  3
2
y  3x  6 x  1
2
Roger Maris, New York Yankees
Outfielder
• “You hit home runs not
by chance but by
preparation.”