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Section 5.1 Quadratic Equations

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**OBJECTIVES Find the greatest common factor (GCF) of numbers.**

Find the GCF of terms.

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**OBJECTIVES Factor out the GCF.**

D Factor a four-term expression by grouping.

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**Greatest Common Factor (GCF)**

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**

a is the greatest integer that divides each coefficient.

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**DEFINITION GCF of a Polynomial**

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

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**Section 5.1 Exercise #2 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Section 5.1 Exercise #5 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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Section 5.2 Quadratic Equations

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OBJECTIVES A Factor trinomials of the form

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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. 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. 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. If c is negative, one integer must be negative and one positive.

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**Section 5.2 Exercise #6 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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Section 5.3 Quadratic Equations

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OBJECTIVES A Use the ac test to determine whether

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OBJECTIVES B

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OBJECTIVES C

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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**

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

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**PROCEDURE Factoring by FOIL**

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**

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

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**Section 5.3 Exercise #8 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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Section 5.4 Quadratic Equations

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OBJECTIVES A Recognize the square of a binomial (a perfect square trinomial).

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OBJECTIVES B Factor a perfect square trinomial.

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OBJECTIVES C Factor the difference of two squares.

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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 5.4 Exercise #11 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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**Section 5.4 Exercise #13 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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Section 5.5 Quadratic Equations

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OBJECTIVES A Factor the sum or difference of two cubes.

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OBJECTIVES B Factor a polynomial by using the general factoring strategy.

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OBJECTIVES C Factor expressions whose leading coefficient is –1.

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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 Factor out all common factors.

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**PROCEDURE General Factoring Strategy**

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 2–A2) and factor it. Note: X 2+A2 is not factorable.

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**PROCEDURE General Factoring Strategy Two terms and cubed:**

Look for the sum of two cubes (X 3+A3) 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**

Let’s work Exercise #19 from Section 5.1

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**Section 5.5 Exercise #15 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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**Section 5.5 Exercise #17 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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**Section 5.5 Exercise #20 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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**Factor out GCF (3) Terms (2) Terms (4) Terms**

Difference of Squares Sum/Difference of Cubes Perfect Square Trinomial (x2 + bx + c) (ax2 + bx + c) Grouping Factoring Strategy Flow Chart

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Section 5.6 Quadratic Equations

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OBJECTIVES A Solve quadratic equations by factoring.

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DEFINITION Quadratic Equation in Standard Form

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**PROCEDURE Solving Quadratics by Factoring**

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

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**PROCEDURE Solving Quadratics by Factoring**

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

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**PROCEDURE Solving Quadratics by Factoring**

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

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**PROCEDURE Solving Quadratics by Factoring**

Solve each of the resulting equations.

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**PROCEDURE Solving Quadratics by Factoring**

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

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**Section 5.6 Exercise #24 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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Solve. or

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Solve. or

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Section 5.7 Quadratic Equations

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OBJECTIVES A Integer problems. B Area and perimeter problems.

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**OBJECTIVES Problems involving the Pythagorean Theorem.**

D Motion problems.

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**NOTE Terminology Notation Examples: 3,4; – 6,–5 2 consecutive integers**

n, n+1 Examples: 3,4; – 6,–5

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**NOTE Terminology Notation 3 consecutive integers n, n+1, n+2**

Examples: 7, 8, 9; – 4,– 3,– 2

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**NOTE Terminology Notation Examples: 8,10; – 6,– 4**

2 consecutive even integers n, n +2 Examples: 8,10; – 6,– 4

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**NOTE Terminology Notation Examples: 13,15; – 21,– 19**

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 5.7 Exercise #26 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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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 5.7 Exercise #29 Chapter 5 Factoring**

Let’s work Exercise #19 from Section 5.1

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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? 10

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