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: 2.8.11.N. Solve quadratic equations.

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)

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

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

Example 1 – Factor by Trial & Error Factor x2 + 5x + 6.

Example 1b Factor x2 – 7x – 30.

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

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

Example 2 – Factor and check by graphing Factor 6x2 + 11x + 3. Check by graphing.

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

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 60x – 1x ≠ 11x -3x + 20x ≠ 11x 3x – 20x ≠ 11x (3x + 2)(x – 10) (3x – 2)(x + 10) (3x + 10)(x – 2) (3x – 10)(x + 2) -30x + 2x ≠ 11x 30x – 2x ≠ 11x -6x +10x ≠ 11x 6x – 10x ≠ 11x (3x + 4)(x – 5) (3x – 4)(x + 5) (3x + 5)(x – 4) (3x – 5)(x + 4) -15x + 4x ≠ 11x 15x – 4x = 11x -12x + 5x ≠ 11x 12x – 5x ≠ 11x

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

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

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

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.

Example 1

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

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

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.

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