CCGPS Analytical Geometry
Quadratics
Factoring PACKET
Name ______________________________________
GCF stands for Greatest Common Factor. Find the GCF of each set of terms.
1) 5, 25, 100
GCF = ______
2) 16, 48, 100
GCF = _______
3) 2x, 10xy, 20x2
GCF = ______
4) 10xy, 15x2y, 30xy2
GCF = _______
Now let’s try them with quadratics (or higher degree polynomials) Examples:
a) 3x2 + 9x + 12
b) 12x2 + 15x
GCF = 3
c) 25x2y – 35xy2
GCF = 3x
(divide each term by 3)
GCF = 5xy
(divide each term by 3x)
3(x2 + 3x + 4)
3x(4x + 5)
(divide each term by 5xy)
5xy(5x – 7y)
Practice: Factor each expression by taking out the GCF.
5) 5x2 + 30x = ___________________
6) 12x2 – 20 = ________________________
7) 36y4 + 9y2 = ___________________
8) 14w3 – 35w = _______________________
Factoring is done to assist us in solving quadratic equations. We use the
Zero Property (anything times zero = 0) and our factored expression to solve.
Examples:
a) 2x2 + 4x = 0
2x(x + 2) = 0
b) 12x2 – 32x = 0
4x(3x – 8) = 0
c) 15x4 – 60x2 = 0
15x2(x2 – 4) = 0
(set each term = 0 and solve)
2x = 0
x+2 = 0
x = 0 & x = -2
4x = 0 3x – 8 = 0
15x2 = 0
x = 0 & x = 8/3
x = 0 & x = ±2
Solve each of the following by factoring out the GCF:
9) 6x2 + 18x = 0
10) 4y2 – 24y = 0
11) 22x2 + 88x = 0
12) 18y4 – 36y2 = 0
13) 8x4 + 64x2 = 0
14) 100y2 – 25y = 0
15) 10x2 + 40x = 0
16) 26y4 – 52y2 = 0
x2 – 4 = 0
Review: Multiply the binomials together:
4) (x + 8)(x + 10) =
5) (x – 7)(x – 8) =
6) (x + 2)(x + 2) =
7) (x – 5)(x + 8) =
8) (x + 11)(x – 10) =
9) (x – 9)(x + 5) =
Factor the trinomials then solve:
10) x2 + 6x + 8 = 0
11) x2 + 6x + 5 = 0
12) x2 + 6x + 9 = 0
13) x2 + 10x + 25 = 0
14) x2 + 10x + 21 = 0
15) x2 + 10x + 9 = 0
16) x2 – 8x + 12 = 0
17) x2 – 8x + 7 = 0
18) x2 – 8x + 16 = 0
19) x2 – 11x +10 = 0
20) x2 – 11x +30 = 0
21) x2 – 11x +28 = 0
22) x2 + 5x – 14 = 0
23) x2 + 5x – 6 = 0
24) x2 + 5x – 36 = 0
Factor out a GCF, then factor and solve:
31) 2x2 + 20x + 48 = 0
32) 3x2 + 24x + 36 = 0
33) 5x2 + 25x + 30 = 0
34) 4x2 – 16x + 16 = 0
35) 2x2 – 14x + 12 = 0
36) 7x2 – 21x + 14 = 0
37) 4x2 + 24x – 28 = 0
38) 2x2 + 20x – 48 = 0
39) 4x2 + 16x – 84 = 0
40) 5x2 + 5x – 150 = 0
41) 3x2 + 6x – 45 = 0
42) 2x2 + 22x – 52 = 0
Multiply each of the following:
1) (x + 5)(x – 5) =
2) (x + 9)(x – 9) =
3) (x – 6)(x + 6) =
7) (2x + 1)(2x – 1) =
8) (4x + 2)(4x – 2) =
9) (5x – 3)(5x + 3) =
* In each of the above products, the middle terms cancel each other out. The products end up
being a “Difference of Squares.” Use this process to factor and solve the following:
13) x2 – 9 = 0
14) x2 – 144 = 0
15) x2 – 25 = 0
16) x2 – 121 = 0
17) x2 – 49 = 0
18) x2 – 1 = 0
19) 4x2 – 16 = 0
20) 9x2 – 64 = 0
21) 25x2 – 1 = 0
Multiply each of the following:
25) (x + 4)(x + 4) =
26) (x + 7)(x + 7) =
27) (x + 12)(x + 12) =
31) (x + 9)2 =
32) (x – 4)2 =
33) (x – 3)2 =
* Look at the above products. How does the middle term relate to the constant term?
(look closely as it may not be evident at first glance – there is a relationship)
Factor and solve each of the following using the above pattern. (these are all perfect squares)
34) x2 + 6x + 9 = 0
35) x2 + 4x + 4 = 0
36) x2 + 10x + 25 = 0
37) x2 + 14x + 49 = 0
38) x2 + 2x + 1 = 0
39) x2 + 16x + 64 = 0
40) x2 + 5x +
25
4
=0
41) x2 + 9x +
CCGPS Honors Geometry
Quadratics
Factoring Wkst #4
81
4
=0
9
42) x2 + 3x + 4 = 0
Name _______________________________________
Date __________________________ Block _______
Review: Factor and Solve
1) x2 + 6x + 8 = 0
2) x2 - 8x + 15 = 0
3) x2 + 12x – 28 = 0
*When a quadratic equation is present, you must move everything over to the same side so the equation = 0.
In the following equations, move all terms to the same side, factor and solve.
7) x2 + 8x = -12
8) x2 – 8 = 7x
10) 16x2 = 36
11) x2 - 6x = -9
9) 4x + 21 = x2
13) 4x2 = 24x – 20
14) 3x2 + 30x = - 75
15) 5x2 – 15 = x2 + 10
16) 6x2 – 18x – 24 = 0
17) 7x2 – 14x + 7 = 0
18) 3x2 – 16x – 12 = 0
19) 6x2 – 5x – 4 = 0
20) 12x2 + 11x + 2 = 0
21) 4x2 + 2x – 12 = 0
22) 3x2 + 23x + 14 = 0
23) 20x2 + x – 1 = 0
24) 12x2 + 11x – 15 = 0
Factor and Solve each of the following:
Use the Quadratic Formula to solve the following. Show all your steps!
1) x2 + 7x + 9 = 0
2) 3x2 – 4x + 2 = 0
3) x2 + 5x – 11 = 0
4) 5x2 – 4x – 10 = 0
5) x2 – 9x – 10 = 0
6) 4x2 + 12x – 8 = 0
Multiply each of the following:
7) (x + 8)2 = ______________________
8) (x – 5)2 = ___________________________
9) (x- 4)2 = _______________________
10) (2x + 7)2 = _________________________
* The solutions to #7-10 are all perfect s________________. How do you know?
Fill in with the number that makes each expression a perfect square.
11) x2 + 12x + __________
12) x2 – 6x + ___________
13) x2 + 2x + _____
14) x2 + 5x + ___________
15) x2 – 9x + ___________
16) x2 + 15x + _____
Solve each of the following by COMPLETING THE SQUARE.
17) x2 + 4x = 5
18) x2 + 8x = 2
19) x2 – 2x = - 8
20) x2 + 6x + 2 = 0
21) x2 – 10x + 5 = 0
22) x2 – 12x = - 3
23) 2x2 + 18x + 20 = 0
24) 3x2 + 12x – 6 = 0
25) 5x2 – 25x + 5 = 0
26) 2x2 + 20x – 5 = 0
27) 4x2 + 20x + 1 = 0
28) 3x2 – 9x + 8 = 0
Solve each of the following – using any method. Show all your steps!
29) 9x4 – 1 = 0
30) 5x2 + 30x + 45 = 0
31) 12x2 – 16x = 0
32) x2 - 6x + 8 = 0
33) 2x2 + 5x + 7 = 0
34) 3x2 – 5x + 2 = 0
35) x2 – 7x – 30 = 0
36) x2 = 2x + 120
37) 2x2 + 6x – 20 = 0
38) 7x2 – 14x + 7 = 0
39) 20x2 – 125 = 0
40) 4x4 = 20x2
41) x2 + 10x + 25 = 0
42) 5x2 – 500 = 0
43) 6x2 – 12x + 6 = 0
44) 6x2 = 24x
45) x2 – 9x – 5 = 0
46) 3x2 – 7x + 1 = 0