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Quadratic Equations

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Factor a four-term expression by grouping. DEFINITION. Greatest Common Factor (GCF) ... Hypotenuse. c. Leg b. Section 5.7. Exercise #26. Chapter 5. Factoring ... – PowerPoint PPT presentation

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Title: Quadratic Equations


1
Section 5.1
Quadratic Equations
2
OBJECTIVES
3
OBJECTIVES
4
DEFINITION
Greatest Common Factor (GCF)
The largest common factor of the integers in a
list.
5
PROCEDURE
Finding the Product
4(x y) 4x 4y
5(a 2b) 5a 10b
2x(x 3) 2x2 6x
6
PROCEDURE
Finding the Factors
4x 4y 4(x y)
5a 10b 5(a 2b)
2x2 6x 2x(x 3)
7
DEFINITION
GCF of a Polynomial
  1. a is the greatest integer that divides each
    coefficient.

8
DEFINITION
GCF of a Polynomial
  1. n is the smallest exponent of x in all the terms.

9
Section 5.1Exercise 2
Chapter 5 Factoring
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Section 5.1Exercise 5
Chapter 5 Factoring
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Section 5.2
Quadratic Equations
16
OBJECTIVES
17
RULE
Factoring Rule 1
18
PROCEDURE
Factoring x2 bx c
Find two integers whose product is c and whose
sum is b.
  1. If b and c are positive, both integers must be
    positive.

19
PROCEDURE
Factoring x2 bx c
Find two integers whose product is c and whose
sum is b.
  1. If c is positive and b is negative, both integers
    must be negative.

20
PROCEDURE
Factoring x2 bx c
Find two integers whose product is c and whose
sum is b.
  1. If c is negative, one integer must be negative
    and one positive.

21
Section 5.2Exercise 6
Chapter 5 Factoring
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Section 5.3
Quadratic Equations
26
OBJECTIVES
27
OBJECTIVES

28
OBJECTIVES

29
TEST
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.
30
TEST
ac test
We need two numbers whose product is ac.
The sum of the numbers must be b.
31
PROCEDURE
Factoring by FOIL
Product must be c.
Product must be a.
32
PROCEDURE
Factoring by FOIL
  1. The product of the numbers in the first (F)
    blanks must be a.

33
PROCEDURE
Factoring by FOIL
  1. The coefficients of the outside (O) products and
    the inside (I) products must add up to b.

34
PROCEDURE
Factoring by FOIL
  1. The product of numbers in the last (L) blanks
    must be c.

35
Section 5.3Exercise 8
Chapter 5 Factoring
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Section 5.4
Quadratic Equations
40
OBJECTIVES
41
OBJECTIVES

42
OBJECTIVES

43
RULES
Factoring Rules 2 and 3
PERFECT SQUARE TRINOMIALS
44
RULES
Factoring Rules 2 and 3
PERFECT SQUARE TRINOMIALS
45
RULE
Factoring Rule 4
THE DIFFERENCE OF TWO SQUARES
46
Section 5.4Exercise 11
Chapter 5 Factoring
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Section 5.4Exercise 13
Chapter 5 Factoring
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Section 5.5
Quadratic Equations
55
OBJECTIVES
56
OBJECTIVES

57
OBJECTIVES

58
RULE
Factoring Rule 5
THE SUM OF TWO CUBES.
59
RULE
Factoring Rule 6
THE DIFFERENCE OF TWO CUBES.
60
PROCEDURE
General Factoring Strategy
  1. Factor out all common factors.

61
PROCEDURE
General Factoring Strategy
  1. Look at the number of terms inside the
    parentheses. If there are

Four terms Factor by grouping.
62
PROCEDURE
General Factoring Strategy
Three terms If the expression is a perfect
square trinomial, factor it. Otherwise, use the
ac test to factor.
63
PROCEDURE
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.
64
PROCEDURE
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.
65
PROCEDURE
General Factoring Strategy
Make sure your expression is completely factored.
Check by multiplying the factors you obtain.
66
Section 5.5
Chapter 5 Factoring
67
Section 5.5Exercise 15
Chapter 5 Factoring
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Section 5.5Exercise 17
Chapter 5 Factoring
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Section 5.5Exercise 20
Chapter 5 Factoring
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Section 5.6
Quadratic Equations
82
OBJECTIVES
83
DEFINITION
Quadratic Equation in Standard Form
84
PROCEDURE
Solving Quadratics by Factoring
  1. Perform necessary operations on both sides so
    that right side 0.

85
PROCEDURE
Solving Quadratics by Factoring
  1. Use general factoring strategy to factor the left
    side if necessary.

86
PROCEDURE
Solving Quadratics by Factoring
  1. Use the principle of zero product and make each
    factor on the left equal 0.

87
PROCEDURE
Solving Quadratics by Factoring
  1. Solve each of the resulting equations.

88
PROCEDURE
Solving Quadratics by Factoring
  1. Check results by substituting solutions obtained
    in step 4 in original equation.

89
Section 5.6Exercise 24
Chapter 5 Factoring
90
Solve.
91
Solve.
92
Section 5.7
Quadratic Equations
93
OBJECTIVES
94
OBJECTIVES
95
NOTE
Notation
Terminology
2 consecutive integers
n, n1
Examples 3,4 6,5
96
NOTE
Notation
Terminology
3 consecutive integers
n, n1, n2
Examples 7, 8, 9 4, 3, 2
97
NOTE
Notation
Terminology
2 consecutive even integers
n, n 2
Examples 8,10 6, 4
98
NOTE
Notation
Terminology
2 consecutive odd integers
n, n 2
Examples 13,15 21, 19
99
DEFINITION
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
100
DEFINITION
Pythagorean Theorem
Hypotenuse c
Leg a
Leg b
101
Section 5.7Exercise 26
Chapter 5 Factoring
102
The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
103
The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
104
The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
105
Section 5.7Exercise 29
Chapter 5 Factoring
106
A 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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