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**Section 5.3 Factoring Quadratic Expressions**

Objectives: Factor a quadratic expression. Use factoring to solve a quadratic equation and find the zeros of a quadratic function. Standard: N. Solve quadratic equations.

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**I. Factoring Quadratic Expressions**

To factor an expression containing two or more terms, factor out the greatest common factor (GCF) of the two expressions. Factor each quadratic expression. 1. 3a2 – 12a 2. 3x(4x + 5) – 5(4x + 5)

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Examples 3. 27a2 – 18a 4. 5x(2x + 1) – 2(2x + 1)

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II. Factoring x2+ bx + c. To factor an expression of the form ax2+ bx + c where a = 1 Find two numbers that add to equal And multiply to equal 5 6 8 7 -26 48 -9 -36

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**Example 1 – Factor by Trial & Error**

Factor x2 + 5x + 6.

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Example 1b Factor x2 – 7x – 30.

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Example 1 c and d c. x2 + 9x + 20 d. x2 – 10x – 11

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**II. Factoring ax2+ bx + c. (Using Trial & Error)**

To factor an expression of the form ax2+ bx + c where a > 1 Find all the factors of c Find all the factors of a Place the factors of a in the first position of each set of parentheses Place the factors of c in the second position of each set of parentheses Try combinations of factors so that when doing FOIL the Firsts mult to equal a; the Outer and Inners mult then add to equal b; the Lasts mult to equal c

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**Example 2 – Factor and check by graphing**

Factor 6x2 + 11x Check by graphing.

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Example 2b Factor 3x2 +11x – 20. Check by graphing.

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**Example 2b 3x2 +11x – 20 Guess and Check**

(3x + 1)(x – 20) (3x – 1)(x + 20) (3x + 20)(x – 1) (3x – 20)(x + 1) -60x +1x ≠ 11x x – 1x ≠ 11x x + 20x ≠ 11x 3x – 20x ≠ 11x (3x + 2)(x – 10) (3x – 2)(x + 10) (3x + 10)(x – 2) (3x – 10)(x + 2) -30x + 2x ≠ 11x x – 2x ≠ 11x x +10x ≠ 11x x – 10x ≠ 11x (3x + 4)(x – 5) (3x – 4)(x + 5) (3x + 5)(x – 4) (3x – 5)(x + 4) -15x + 4x ≠ 11x x – 4x = 11x x + 5x ≠ 11x 12x – 5x ≠ 11x

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1. 3x2 + 18 4. x2 – 10x - 24 5. x2 + 4x - 32 2. x – 4x2 6. 3x2 + 7x + 2 3. x2 + 8x + 16 7. 3x2 – 5x - 2

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**Factoring the Difference of 2 Squares**

a2 – b2 = (a + b)(a – b) Factor the following expressions: 1. y2 - 25 2. 9x4 - 49 3. x4 - 16

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**Factoring Perfect Square Trinomials**

a2 + 2ab + b2 = (a + b)2 or a2 – 2ab + b2 = (a – b)2 Factor the following expressions: 4x2 – 24x + 36 These are called a Perfect Square Trinomial because: 9x2 – 36x + 36

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**Zero-Product Property**

A zero of a function f is any number r such that f(r) = 0. Zero-Product Property When multiplying two numbers p and q: If p = 0 then p ● q = 0. If q = 0 then p ● q = 0. An equation in the form of ax2+ bx + c = 0 is called the general form of a quadratic equation. The solutions to this equation are called the zeros and are the locations where the parabola crosses the x-axis.

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Example 1

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**Example 1 c and d c. f(x) = 3x2 – 12x d. g(x) = x2 + 4x - 21**

Use the zero product property to find the zeros of each function. c. f(x) = 3x2 – 12x d. g(x) = x2 + 4x - 21

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Pg. 296 #59, 61, 63 1. 3x2 – 5x = 2 2. 6x2 – 17x = -12 3. 3x2 + 3 = 10

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Writing Activities 2. a. Shannon factored 4x2 – 36x + 81 as (2x + 9)2. Was she correct? Explain. b. Brandon factored 16x2 – 25 as (4x – 5)2. Was he correct? Explain.

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Homework Integrated Algebra II- Section 5.3 Level A Academic Algebra II- Section 5.3 Level B

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