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```					Factoring Quadratic Expressions
LESSON 5-4

Factor each expression.

a. 15x2 + 25x + 100
15x2 + 25x + 100 = 5(3x2) + 5(5x) + 5(20)   Factor out the GCF, 5
= 5(3x2 + 5x + 20)      Rewrite using the
Distributive Property.

b. 8m2 + 4m
8m2 + 4m = 4m(2m) + 4m(1)                   Factor out the GCF, 4m
= 4m(2m + 1)                    Rewrite using the
Distributive Property.

ALGEBRA 2
LESSON 5-4

Factor x2 + 10x + 24.

Step 1: Find factors with product ac and sum b.
Since ac = 24 and b = 10, find positive factors with product 24
and sum 11.
Factors of 24        1, 24     2, 12      3, 8      6, 4
Sum of factors        25        14         11        10

Step 2: Rewrite the term bx using the factors you found. Group the
remaining terms and find the common factors for each group.
x2 + 10x + 24
x2 + 4x + 6x + 24             Rewrite bx : 10x = 4x + 6x.
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x(x + 4) + 6(x + 4)           Find common factors.

ALGEBRA 2
LESSON 5-4

(continued)

Step 3: Rewrite the expression as a product of two binominals.

x(x + 4) + 6(x + 4)

(x + 6)(x + 4)        Rewrite using the Distributive Property.

Check: (x + 6)(x + 4) = x2 + 4x + 6x + 24
= x2 + 10x + 24

ALGEBRA 2
LESSON 5-4

Factor x2 – 14x + 33.

Step 1: Find factors with product ac and sum b.
Since ac = 33 and b = –14, find negative factors with product 33
and sum b.
Factors of 33         –1, –33       –3, –11
Sum of factors         –34           –14

Step 2: Rewrite the term bx using the factors you found. Then find common
factors and rewrite the expression as a product of two binomials.
x2 – 14x + 33
x2 – 3x – 11x + 33         Rewrite bx.
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}

x(x – 3) – 11(x – 3)             Find common factors.
(x – 11)(x – 3)                 Rewrite using the Distributive Property.

ALGEBRA 2
LESSON 5-4

Factor x2 + 3x –28.

Step 1: Find factors with product ac and sum b.
Since ac = –28 and b = 3, find factors 2 with product –28 and
sum 3.
Factors of –28    1, –28   –1, 28   2, –14   –2, 14   4, –7   –4, 7
Sum of factors     –27      27       –12      12       –3       3

Step 2: Since a = 1, you can write binomials using the factors you found.
x2 + 3x – 28
(x – 4)(x + 7)              Use the factors you found.

ALGEBRA 2
LESSON 5-4

Factor 6x2 – 31x + 35.

Step 1: Find factors with product ac and sum b.
Since ac = 210 and b = –31, find negative factors with product
210 and sum –31.
Factors of 210    –1, –210 –2, –105 –3, –70 –5, –42 –10, –21
Sum of factors      –211     –107    –73     –47      –31

Step 2: Rewrite the term bx using the factors you found. Then find common
factors and rewrite the expression as the product of two binomials.
6x2 – 31x + 35
6x2 – 10x – 21x + 35          Rewrite bx.
}
}

2x(3x – 5) – 7(3x – 5)        Find common factors.
(2x – 7)(3x – 5)              Rewrite using the Distributive Property.

ALGEBRA 2
LESSON 5-4

Factor 6x2 + 11x – 35.
Step 1: Find factors with product ac and sum b.
Since ac = 210 and b = 11, find factors with product –210 and
sum 11.
Factors of –210 –1, –210     –1, 210   2, –105 –2, 105    3, –70
Sum of factors    –209        209       –103    103        –67

Factors of –210    –3, 70    5, –42    –5, 42   10, –21   –10, 21
Sum of factors      67        –37       37        –11       11

Step 2: Rewrite the term bx using the factors you found. Then find common
factors and rewrite the expression as the product of two binomials.
6x2 + 11x + 35
6x2 – 10x + 21x – 35        Rewrite bx.
2x(3x – 5) + 7(3x – 5)      Find common factors.
(2x + 7)(3x – 5)            Rewrite using the Distributive Property.

ALGEBRA 2
LESSON 5-4

Factor 100x2 + 180x + 81.
100x2 + 180x + 81 = (10x)2 + 180 + (9)2           Rewrite the first and third
terms as squares.

= (10x)2 + 180 + (9)2   Rewrite the middle term to
verify the perfect square
trinomial pattern.

= (10x + 9)2            a2 + 2ab + b2 = (a + b)2

ALGEBRA 2
LESSON 5-4

A square photo is enclosed in a square frame, as shown in
the diagram. Express the area of the frame (the shaded area) in
completely factored form.

Relate: frame area equals the outer area minus the inner area

Define: Let x = length of side of frame.

Write:     area = x2 – (7)2

= (x + 7)(x – 7)

The area of the frame in factored form is (x + 7)(x – 7) in2.

ALGEBRA 2

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