Quadratic Equations

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

1. a is the greatest integer that divides each
   coefficient.

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DEFINITION
GCF of a Polynomial

1. n is the smallest exponent of x in all the terms.

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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
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OBJECTIVES
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RULE
Factoring Rule 1
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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.

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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.

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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.

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Section 5.2Exercise 6
Chapter 5 Factoring
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Section 5.3
Quadratic Equations
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OBJECTIVES
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OBJECTIVES

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OBJECTIVES

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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.
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TEST
ac test
We need two numbers whose product is ac.
The sum of the numbers must be b.
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PROCEDURE
Factoring by FOIL
Product must be c.
Product must be a.
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PROCEDURE
Factoring by FOIL

1. The product of the numbers in the first (F)
   blanks must be a.

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PROCEDURE
Factoring by FOIL

1. The coefficients of the outside (O) products and
   the inside (I) products must add up to b.

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PROCEDURE
Factoring by FOIL

1. The product of numbers in the last (L) blanks
   must be c.

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Section 5.3Exercise 8
Chapter 5 Factoring
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Section 5.4
Quadratic Equations
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OBJECTIVES
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OBJECTIVES

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OBJECTIVES

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RULES
Factoring Rules 2 and 3
PERFECT SQUARE TRINOMIALS
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RULES
Factoring Rules 2 and 3
PERFECT SQUARE TRINOMIALS
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RULE
Factoring Rule 4
THE DIFFERENCE OF TWO SQUARES
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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
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OBJECTIVES
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OBJECTIVES

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OBJECTIVES

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RULE
Factoring Rule 5
THE SUM OF TWO CUBES.
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RULE
Factoring Rule 6
THE DIFFERENCE OF TWO CUBES.
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PROCEDURE
General Factoring Strategy

1. Factor out all common factors.

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PROCEDURE
General Factoring Strategy

1. Look at the number of terms inside the
   parentheses. If there are

Four terms Factor by grouping.
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PROCEDURE
General Factoring Strategy
Three terms If the expression is a perfect
square trinomial, factor it. Otherwise, use the
ac test to factor.
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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.
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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.
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PROCEDURE
General Factoring Strategy
Make sure your expression is completely factored.
Check by multiplying the factors you obtain.
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Section 5.5
Chapter 5 Factoring
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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
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OBJECTIVES
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DEFINITION
Quadratic Equation in Standard Form
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PROCEDURE
Solving Quadratics by Factoring

1. Perform necessary operations on both sides so
   that right side 0.

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PROCEDURE
Solving Quadratics by Factoring

1. Use general factoring strategy to factor the left
   side if necessary.

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PROCEDURE
Solving Quadratics by Factoring

1. Use the principle of zero product and make each
   factor on the left equal 0.

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PROCEDURE
Solving Quadratics by Factoring

1. Solve each of the resulting equations.

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PROCEDURE
Solving Quadratics by Factoring

1. Check results by substituting solutions obtained
   in step 4 in original equation.

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Section 5.6Exercise 24
Chapter 5 Factoring
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Solve.
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Solve.
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Section 5.7
Quadratic Equations
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OBJECTIVES
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OBJECTIVES
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NOTE
Notation
Terminology
2 consecutive integers
n, n1
Examples 3,4 6,5
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NOTE
Notation
Terminology
3 consecutive integers
n, n1, n2
Examples 7, 8, 9 4, 3, 2
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NOTE
Notation
Terminology
2 consecutive even integers
n, n 2
Examples 8,10 6, 4
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NOTE
Notation
Terminology
2 consecutive odd integers
n, n 2
Examples 13,15 21, 19
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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
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DEFINITION
Pythagorean Theorem
Hypotenuse c
Leg a
Leg b
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Section 5.7Exercise 26
Chapter 5 Factoring
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The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
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The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
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The product of two consecutive odd integers is 13
more than 10 times the larger of the two
integers. Find the integers.
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Section 5.7Exercise 29
Chapter 5 Factoring
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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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