Section 1.5 Quadratic Equations. Solving Quadratic Equations by Factoring.
Section 5.3 Factoring Quadratic Expressions
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Transcript of Section 5.3 Factoring Quadratic Expressions
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Section 5.3Factoring 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.
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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 thatadd to equal And multiply to equal
5 6
8 7
-26 48
-9 -36
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Factor by Trial & Error
Factor x2 + 5x + 6.
Factor x2 – 7x – 30.
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Factor x2 + 9x + 20
Factor x2 – 10x – 11
Factor by Trial & Error
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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 + 3. Check by graphing.
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Example 2b
Factor 3x2 +11x – 20. Check by graphing.
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Example 2b3x2 +11x – 20Guess 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
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1. 3x2 + 18
2. x – 4x2
3. x2 + 8x + 16
5. x2 + 4x - 32
4. x2 – 10x - 24
6. 3x2 + 7x + 2
7. 3x2 – 5x - 2
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Factoring the Difference of 2 Squares
a2 – b2 = (a + b)(a – b)
Factor the following expressions:
3. x4 - 16
1. y2 - 25 2. 9x4 - 49
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Factoring Perfect Square Trinomials
bca )(2
a2 + 2ab + b2 = (a + b)2 or a2 – 2ab + b2 = (a – b)2
4x2 – 24x + 36
Factor the following expressions:
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 PropertyWhen 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.
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Factor, use zero product property
1. 3x2 – 5x = 2
3. 3x2 + 3 = 10
2. 6x2 – 17x = -12
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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
Honors Algebra II- Section 5.3 Level B