Transcript Intermediate Algebra Chapter 9
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
ax
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
2
c then x
x c or
x
c c
Sample problem
x
2 40
x
40
x
x
2 10
Sample problem 2 5
x
2 62 5
x
2
x
2 60 12
x
x
12 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
x
x
3 Sample problem 3 2 16
x x
3 16 4
x
3 4
or x x
1
or x
7
7 2
x x
3 2 Sample problem 4
x
3 2 0 7 2
x
3 2 7 25 7 25 5 7
i x
3 2 7
i
10 1.5
0.26
i
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
• 3. Complete the square by adding the square of half of the linear coefficient to both sides.
k
• 4. Use square root property • 5. Simplify
Sample Problem
x
2 8
x x
11 0
x
Sample Problem complete the square 2 2 5
x x
5 2 29
Sample problem complete the square #3 3
x
2 7
x
10 4
x
7 6 23
i
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.
3
x
2 8
x x
4 3 3 5
i
0 Quadratic Formula • For all a,b, and c that are real numbers and a is not equal to zero
x
b
2 4
ac
2
a
2
x
Sample problem quadratic formula #1 2 9
x
0 1 2
Sample problem quadratic formula #2
x
2 12
x
0
x
2 10
Sample problem quadratic formula #3 3
x
2
x
8
x
4 3 3 5
i
0
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
b
2 4
ac
Negative Zero
root) – complex conjugates – one rational solution (double
Positive
– Perfect square – 2 rational solutions – Not perfect square – 2 irrational solutions
Sum of Roots
r
1 2
b a
Product of Roots
r r
1 2
c 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
2 2
x
2,4 0
Sample Problem quadric inequalities #2 6
x
2 , 1 2 2 3 , 2
Sample Problem quadratic inequalities #3
x
2
R
6
x
0
Sample Problem quadratic
x x
inequalities #4 4 1 0
Sample Problem quadratic inequalities #5
x
3 2 2,3
x
2 3 5,
0
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
h
2
k
Find Vertex • x coordinate is
b
2
a
• y coordinate is
f
b
2
a
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
y y
Sample Problems - graph
x
2 6
x
5
x
2
x
3
y
3
x
2
6
x
1
Roger Maris, New York Yankees Outfielder •“You hit home runs not by chance but by preparation.”