Title: Quadratic Equations
1Section 5.1
Quadratic Equations
2OBJECTIVES
3OBJECTIVES
4DEFINITION
Greatest Common Factor (GCF)
The largest common factor of the integers in a
list.
5PROCEDURE
Finding the Product
4(x y) 4x 4y
5(a 2b) 5a 10b
2x(x 3) 2x2 6x
6PROCEDURE
Finding the Factors
4x 4y 4(x y)
5a 10b 5(a 2b)
2x2 6x 2x(x 3)
7DEFINITION
GCF of a Polynomial
- a is the greatest integer that divides each
coefficient.
8DEFINITION
GCF of a Polynomial
- n is the smallest exponent of x in all the terms.
9Section 5.1Exercise 2
Chapter 5 Factoring
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11Section 5.1Exercise 5
Chapter 5 Factoring
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15Section 5.2
Quadratic Equations
16OBJECTIVES
17RULE
Factoring Rule 1
18PROCEDURE
Factoring x2 bx c
Find two integers whose product is c and whose
sum is b.
- If b and c are positive, both integers must be
positive.
19PROCEDURE
Factoring x2 bx c
Find two integers whose product is c and whose
sum is b.
- If c is positive and b is negative, both integers
must be negative.
20PROCEDURE
Factoring x2 bx c
Find two integers whose product is c and whose
sum is b.
- If c is negative, one integer must be negative
and one positive.
21Section 5.2Exercise 6
Chapter 5 Factoring
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25Section 5.3
Quadratic Equations
26OBJECTIVES
27OBJECTIVES
28OBJECTIVES
29TEST
ac test for ax2 bx c
A trinomial of the form ax2 bx c is
factorable if there are two integers with product
ac and sum b.
30TEST
ac test
We need two numbers whose product is ac.
The sum of the numbers must be b.
31PROCEDURE
Factoring by FOIL
Product must be c.
Product must be a.
32PROCEDURE
Factoring by FOIL
- The product of the numbers in the first (F)
blanks must be a.
33PROCEDURE
Factoring by FOIL
- The coefficients of the outside (O) products and
the inside (I) products must add up to b.
34PROCEDURE
Factoring by FOIL
- The product of numbers in the last (L) blanks
must be c.
35Section 5.3Exercise 8
Chapter 5 Factoring
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39Section 5.4
Quadratic Equations
40OBJECTIVES
41OBJECTIVES
42OBJECTIVES
43RULES
Factoring Rules 2 and 3
PERFECT SQUARE TRINOMIALS
44RULES
Factoring Rules 2 and 3
PERFECT SQUARE TRINOMIALS
45RULE
Factoring Rule 4
THE DIFFERENCE OF TWO SQUARES
46Section 5.4Exercise 11
Chapter 5 Factoring
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50Section 5.4Exercise 13
Chapter 5 Factoring
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54Section 5.5
Quadratic Equations
55OBJECTIVES
56OBJECTIVES
57OBJECTIVES
58RULE
Factoring Rule 5
THE SUM OF TWO CUBES.
59RULE
Factoring Rule 6
THE DIFFERENCE OF TWO CUBES.
60PROCEDURE
General Factoring Strategy
- Factor out all common factors.
61PROCEDURE
General Factoring Strategy
- Look at the number of terms inside the
parentheses. If there are
Four terms Factor by grouping.
62PROCEDURE
General Factoring Strategy
Three terms If the expression is a perfect
square trinomial, factor it. Otherwise, use the
ac test to factor.
63PROCEDURE
General Factoring Strategy
Two terms and squared Look at the
difference of two squares (X 2A2) and factor it.
Note X 2A2 is not factorable.
64PROCEDURE
General Factoring Strategy
Two terms and cubed Look for the sum of two
cubes (X 3A3) or the difference of two cubes (X
3-A3) and factor it.
65PROCEDURE
General Factoring Strategy
Make sure your expression is completely factored.
Check by multiplying the factors you obtain.
66Section 5.5
Chapter 5 Factoring
67Section 5.5Exercise 15
Chapter 5 Factoring
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71Section 5.5Exercise 17
Chapter 5 Factoring
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77Section 5.5Exercise 20
Chapter 5 Factoring
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81Section 5.6
Quadratic Equations
82OBJECTIVES
83DEFINITION
Quadratic Equation in Standard Form
84PROCEDURE
Solving Quadratics by Factoring
- Perform necessary operations on both sides so
that right side 0.
85PROCEDURE
Solving Quadratics by Factoring
- Use general factoring strategy to factor the left
side if necessary.
86PROCEDURE
Solving Quadratics by Factoring
- Use the principle of zero product and make each
factor on the left equal 0.
87PROCEDURE
Solving Quadratics by Factoring
- Solve each of the resulting equations.
88PROCEDURE
Solving Quadratics by Factoring
- Check results by substituting solutions obtained
in step 4 in original equation.
89Section 5.6Exercise 24
Chapter 5 Factoring
90Solve.
91Solve.
92Section 5.7
Quadratic Equations
93OBJECTIVES
94OBJECTIVES
95NOTE
Notation
Terminology
2 consecutive integers
n, n1
Examples 3,4 6,5
96NOTE
Notation
Terminology
3 consecutive integers
n, n1, n2
Examples 7, 8, 9 4, 3, 2
97NOTE
Notation
Terminology
2 consecutive even integers
n, n 2
Examples 8,10 6, 4
98NOTE
Notation
Terminology
2 consecutive odd integers
n, n 2
Examples 13,15 21, 19
99DEFINITION
Pythagorean Theorem
If the longest side of a right triangle is of
length c and the other two sides are of length a
and b, then
100DEFINITION
Pythagorean Theorem
Hypotenuse c
Leg a
Leg b
101Section 5.7Exercise 26
Chapter 5 Factoring
102The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
103The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
104The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
105Section 5.7Exercise 29
Chapter 5 Factoring
106A rectangular 10-inch television screen (measured
diagonally) is 2 inches wider than it is high.
What are the dimensions of the screen?
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