NAME ______________________________________
Factor each expression using the GCF.
1. __________ 4 x  16
2
2. __________ 5m  35
2
3. __________ 2 x  4
2
4. __________ 5r  10 r
5. __________ 3 p 2  12 p
6. __________ 5am  5an
8. __________ 4 x 4  24 x 2
9. __________ k 5  k 2
2
4
11. __________ 3x  21x
2
12. __________ 3x  6 x
7. __________
3x 2  4 y
3
5
10. __________ 2t  130 t
13. ____________________ 7 x 2  28 x  14
14. ____________________
100  25 s 5  50 s
15. ____________________ x 9  x 2
16. ____________________
3m3  9m 2  3m
17. ____________________ 4n 3  16 n 2  8n
18. ____________________
6w 4  8w 2  12w
19. ____________________ 7 x 3  21x 4
20. ____________________
4a 3  20 a 2  8a
Factor each expression using the GCF.
KEY




4 x  4
1. __________ 4 x  16
5 m2  7
2
2. __________ 5m  35
2 x2  2
2
3. __________ 2 x  4
5r r  2 5r 2  10 r
4. __________
3 p p  4
5. __________ 3 p 2  12 p
5am  n 5am  5an
6. __________
3x 2  4 y
7. __________
x x  6 4 x 4  24 x 2
8. 4
__________

3x 2  4 y

2t 3 1  65t 2
3
5
10. __________ 2t  130 t

2




3x2 1  7x2
3x 2  21x 4
15. ____________________



x2 x7  1
11. __________
7 x2  4x  2
13. ____________________ 7 x 2  28 x  14

2



k2 k3  1
9. __________ k 5  k 2
3 x x  2
2
12. __________ 3x  6 x
x9  x 2


25 4  s 5  2 s
14. ____________________


3m m 2  3m  1
16. ____________________

2w 3w 2  4w  6

4n n 2  4n  2
17. ____________________ 4n 3  16 n 2  8n
18. ____________________
7 x 3 1  3 x 
19. ____________________ 7 x 3  21x 4
4a a 2  5a  2
20. ____________________


100  25 s 5  50 s
3m3  9m 2  3m
6w 4  8w 2  12w
4a 3  20 a 2  8a
NAME ______________________________________
The Riddle Game
Find 2 numbers whose product is the 1st number and their sum is the second number.
1. __________
14 , 9
21. __________ 25 , 10
2. __________
9,6
22. __________ -45 , 4
3. __________
10 , 11
23. __________ -8 , -7
4. __________
24 , 11
24. __________ 9 , -6
5. __________
50 , 27
25. __________ -30 , 7
6. __________
2,3
26. __________ -18 , -7
7. __________
16 , 8
27. __________ -18 , 3
8. __________
16 , 10
28. __________ -18 , 17
9. __________
48 , 19
29. __________ 6, 5
10. __________
5 , -6
30. __________ -55 , -6
11. __________
51 , 20
31. __________ -2 , 1
12. __________
-20 , 1
32. __________ 100 , 101
13. __________
-20 , -1
33. __________ 28 , 16
14. __________
20 , -9
34. __________ 28 , -11
15. __________
-12, 4
35. __________ -15 , -2
16. __________
-80 , -2
36. __________ -8 , -2
17. __________
40 , -13
37. __________ 21 , 10
18. __________
-7 , -6
38. __________ 30 , 11
19. __________
-4 , 0
39. __________ -70 , 33
20. __________
12 , -8
40. __________ 110, 21
The Riddle Game
Find 2 numbers whose product is the 1st number and their sum is the second number.
KEY
2, 7
1. __________
14 , 9
5, 5
21. __________
25 , 10
3, 3
2. __________
9,6
-5, 9
22. __________
-45 , 4
1, 10
3. __________
10 , 11
-8, 1
23. __________
-8 , -7
3, 8
4. __________
24 , 11
-3, -3
24. __________
9 , -6
2, 25
5. __________
50 , 27
10, -3
25. __________
-30 , 7
1, 2
6. __________
2,3
-9, 2
26. __________
-18 , -7
4, 4
7. __________
16 , 8
6, -3
27. __________ -18 , 3
2, 8
8. __________
16 , 10
18, -1
28. __________ -18 , 17
3, 16
9. __________
48 , 19
2, 3
29. __________
6, 5
-1, -5
10. __________
5 , -6
-11, 5
30. __________
-55 , -6
3, 17
11. __________
51 , 20
2, -1
31. __________
-2 , 1
5, -4
12. __________
-20 , 1
100, 1
32. __________
100 , 101
-5, 4
13. __________
-20 , -1
2, 14
33. __________
28 , 16
-4, -5
14. __________
20 , -9
-4, -7
34. __________
28 , -11
6, -2
15. __________
-12, 4
-5, 3
35. __________
-15 , -2
-10, 8
16. __________
-80 , -2
-4, 2
36. __________ -8 , -2
-5, -8
17. __________
40 , -13
3, 7
37. __________
21 , 10
-7, 1
18. __________
-7 , -6
5, 6
38. __________
30 , 11
-2, 2
19. __________
-4 , 0
35, -2
39. __________
-70 , 33
-2, -6
20. __________
12 , -8
10, 11
40. __________
110, 21
NAME ______________________________________
Factor each expression using the Riddle Game.
1. __________________________ x 2  9 x  14
2. __________________________ x 2  4 x  45
3. __________________________ x 2  11x  10
2
4. __________________________ x  6 x  9
2
5. __________________________ x  2 x  15
2
6. __________________________ x  x  2
7. __________________________ x 2  6 x  7
2
8. __________________________ x  16 x  28
9. __________________________ x 2  2 x  80
10. __________________________ x 2  5x  6
2
11. __________________________ x  11x  30
2
12. __________________________ x  3x  18
2
13. __________________________ x  4 x  21
2
14. __________________________ x  13 x  40
15. __________________________ x  x  20
2
Factor each expression using the Riddle Game.
KEY
x  2x  7
1. __________________________
x 2  9 x  14
 x  9 x  5
2. __________________________
x 2  4 x  45
x  10x  1
x 2  11x  10
3. __________________________
x  3x  3
x 2  6x  9
4. __________________________
 x  5 x  3
x 2  2 x  15
5. __________________________
 x  2 x  1
x2  x  2
6. __________________________
 x  7 x  1
7. __________________________
x 2  6x  7
 x  14 x  2
x 2  16 x  28
8. __________________________
 x  10 x  8
9. __________________________
x 2  2 x  80
 x  6 x  1
x 2  5x  6
10. __________________________
 x  5 x  6
x 2  11x  30
11. __________________________
 x  6 x  3
x 2  3x  18
12. __________________________
 x  7 x  3
x 2  4 x  21
13. __________________________
 x  5 x  8
x 2  13 x  40
14. __________________________
 x  5 x  4
x  x  20
15. __________________________
2
NAME ______________________________________
Factor each difference of 2 squares.
2
1. ___________________ d  4
6
10. ___________________ x  1
2. ___________________ x 2  64
11. ___________________ 9 z 2  49
3. ___________________ d 2  4
2
2
12. ___________________ a  4b
4. ___________________ y 4  25
2
2
13. ___________________ x  y
5. ___________________ r 2  36
2
2
14. ___________________ d  9y
2
6. ___________________ w  144
2
15. ___________________ 4 x  9
2
7. ___________________ b  1
4
2
16. ___________________ 81x  16 y
2
8. ___________________ f  20
2
2
17. ___________________ g  121y
2
9. ___________________ q  100
2
18. ___________________ 16 x  25
Factor each difference of 2 squares.
KEY



d  2d  2
d2 4
1. ___________________
x3  1 x3  1
6
10. ___________________ x  1
 x  8 x  8
2. ___________________
x 2  64
3z  73z  7 9 z 2  49
11. ___________________
d2  4
3. ___________________
d2 4
(prime)
a  2ba  2b a 2  4b 2
12. ___________________



y2  5 y2  5
4. ___________________
y 4  25
r  6r  6
 x  y  x  y  x 2  y 2
13. ___________________
5. ___________________ r 2  36
2
d  3 yd  3 y 2
14. ___________________ d  9y
w  12w  12
2
6. ___________________ w  144
2 x  32 x  3
2
15. ___________________ 4 x  9
b  1b  1
b 1
7. ___________________
2


f  20
f 2  20
8. ___________________
(prime)

2


q  10 q  10
2
9. ___________________ q  100
9 x
2

 4 y 9x2  4 y

4
2
16. ___________________ 81x  16 y
2
g  11 yg  11 y 2
17. ___________________ g  121y
4 x  54 x  5
2
18. ___________________ 16 x  25
NAME ______________________________________
Combination Factoring
Steps for factoring:
1. Factor out GCF
2. Factor trinomial (riddle game)
3. Factor difference of 2 squares
Factor completely.
1. _____________________ 2 x 2  8
2
2. _____________________ 3x  15 x  18
2
3. _____________________ 2w  32
2
4. _____________________ 5 x  45
5. _____________________ 10 x 2  10
6. _____________________  2 x 2  8x  42
7. _____________________ 3 y 6  75
2
8. _____________________ 8x  80 x  200
2
9. _____________________ 4t  28t  48
2
10. _____________________ 5r  25r  20
Combination Factoring
Steps for factoring:
4. Factor out GCF
5. Factor trinomial (riddle game)
6. Factor difference of 2 squares
Factor completely.
KEY
2 x  2 x  2
2x 2  8
1. _____________________
3 x  2 x  3
3x 2  15 x  18
2. _____________________



2 w 4 w 4
2w 2  32
3. _____________________
5 x  3 x  3
5 x 2  45
4. _____________________
10 x  1 x  1
5. _____________________
10 x 2  10
 2 x  7 x  3
6. _____________________
 2 x 2  8x  42



3 y3  5 y3  5
3 y 6  75
7. _____________________
8 x  5 x  5
8x 2  80 x  200
8. _____________________
4t  4t  3
2
9. _____________________ 4t  28t  48
5r  1r  4
2
10. _____________________ 5r  25r  20
NAME ______________________________________
SOLVE EQUATIONS BY FACTORING
To solve equations by factoring:
a. Set equation equal to zero
b. Factor
c. Use the Zero Product Property to find solutions
Zero Product Property (Zero Guarantee): If ab = 0, then a = 0 or b = 0
I.
II.
Use the Zero Product Property to find solutions.
1.
__________
x  3x  7  0
4. __________
 y  2 y  3  0
2.
_________
x x  6   0
5. __________
c  7c  7  0
3.
_________
3r r  1r  9  0
6. __________
x  6x  2x  5  0
Solve each equation by factoring.
7.
2
_________ 2 x  18
11. _________
y 2  64  0
8.
_________ x  9 x  20  0
12. _________ r  5r  6
9.
2
_________ 3w  6w  24
2
13. _________ x  81
2
2
10. _________ 4a  4
2
2
14. _________ y  7 y  10
SOLVE EQUATIONS BY FACTORING (CONT)
Solve each equation by factoring:
2
15. _________ 9 x  4  0
2
22. _________ t  9  6t
16._________ x 2  x  2
23. _________ 2 x 2  30  16 x
2
17._________  3q  9q  30  0
24. _________ x  16 x  15
2
18._________
5 x 2  5 x  30  0
25. _________ x  12 x  27  0
19._________
h 2  14h  40
2
26. _________ 2 x  8 x  42  0
20._________
x3  x  0
2
27. _________ d  13d  22  0
21._________
10 p 2  70 p  100  0
2
28. _________ f  4 f  12
2
NAME ______________________________________
SOLVE EQUATIONS BY FACTORING
KEY
To solve equations by factoring:
d. Set equation equal to zero
e. Factor
f. Use the Zero Product Property to find solutions
Zero Product Property (Zero Guarantee): If ab = 0, then a = 0 or b = 0
I.
II.
Use the Zero Product Property to find solutions.
3,7
x  3x  7  0
4. __________
0,6
_________
 
x x  6   0
 7,7
5. __________

3r r  1r  9  0
 6,2,5
6. __________
1.
__________
2.
3.
 1,0,9
_________

 2,3
 y  2 y  3  0

c  7c  7  0


 x  6x  2x  5  0
Solve each equation by factoring.
 3,3
7.
_________ 2 x  18
8.
 5,-4 x  9 x  20  0
_________
9.
4,-2
3w 2  6w  24
_________






2
2
2
- 1,1
10. _________ 4a  4
- 8,8
11. _________






y 2  64  0
- 1,6 r  5r  6
12. _________
2
- 9,9 x 2  81
13. _________
2
- 5,-2
14. _________ y  7 y  10
SOLVE EQUATIONS BY FACTORING (CONT)
KEY
Solve each equation by factoring:
 2 2
- ,  2
 3 3  9x  4  0
15. _________




- 2,1
16._________ x 2  x  2
- 5,2
 3q 2  9q  30  0
17._________
- 3,2
18._________

3
t 2  9  6t
22. _________
 
2
3,5
23. _________ 2 x  30  16 x


- 15,-1 x  16 x  15
24. _________
3,9
2
5 x 2  5 x  30  0
25. _________ x  12 x  27  0


h 2  14h  40
- 3,7
2 x 2  8 x  42  0
26. _________


x3  x  0
2
27. _________ d  13d  22  0
10 p 2  70 p  100  0
28. _________
- 10,-4
19._________
- 1,0,1
20._________
- 5,-2
21._________

2

- 11,-2
- 2,6
f 2  4 f  12
NAME ___________________________________
x
QUADRATIC FORMULA
I.
II.
 b  b 2  4ac
2a
Identify a, b, and c. Find the discriminant (b2 - 4ac) and name the number of solutions
(0, 1 or 2) for each equation.
1.
x 2  4x  3  0
4.
x 2  3x  10  0
2.
x 2  2x  1  0
5.
x 2  5x  2  0
3.
 x 2  4x  2
6.
7 x 2  2x  1  0
Use the quadratic formula to solve each quadratic equation.
If necessary, round answers to the nearest hundredth.
7.
2x 2  4x  7  0
9.
8.
3x 2  4 x  30
10.
2x 2  5x  3  0
5 x 2  13x  1  0
QUADRATIC FORMULA (CONT)
Use the quadratic formula to solve each quadratic equation.
If necessary, round answers to the nearest hundredth.
11.
5 x 2  16 x  84  0
15.
7 x 2  100 x  4  0
12.
 3x 2  5 x  2  0
16.
4 x 2  12 x  9  0
13.
2 x 2  9 x  221  0
17.
x 2  4 x  10  0
14.
x 2  6x  7  0
18.
x 2  2x  2  0
x
 b  b 2  4ac
2a
QUADRATIC FORMULA (CONT)
x
III.
Use the formula ho  16t 2  vt  hi to solve the following.
If necessary, round answers to the nearest hundredth.
 b  b 2  4ac
2a
19.
Brandy tosses a ball upward with a starting velocity of 10 feet per second. She released
the ball from a height of 3 feet. If the ball is not caught, how long will the ball be in the
air?
20.
Jerry threw a ball upward with a starting velocity of 30 feet per second. The ball was 6
feet high when it left his hand. How many seconds (t) did it take for the ball to hit the
ground?
21.
A soccer ball bounces straight up into the air off the head of a soccer player at an
altitude of 5 feet with an initial velocity of 30 feet per second. How long will it take for
the ball to hit the ground?
22.
Benny the frog jumps from one lily pad to the next lily pad. If he jumps with an initial
velocity of 12 feet per second, how long will he take to reach the next lily pad?
23.
Michael shot a 3 point shot. He launched the basketball at 15 feet per second. The ball
left his hand at an altitude of 7 feet. How long would it take the ball to get to the rim
which is 10 feet above the ground?
NAME ___________________________________
x
QUADRATIC FORMULA
 b  b 2  4ac
2a
KEY
I.
Identify a, b, and c. Find the discriminant (b2 - 4ac) and name the number of solutions
(0, 1 or 2) for each equation.
1.
x 2  4x  3  0
4.
a  1, b  3, c  10
b 2  4ac  31
Solutions  0
a  1, b  4, c  3
b 2  4ac  4
Solutions  2
2.
x 2  2x  1  0
5.
a  1, b  2, c  1
b 2  4ac  0
Solutions  1
3.
 x 2  4x  2
x 2  5x  2  0
a  1, b  5, c  2
b 2  4ac  17
Solutions  2
6.
 x 2  4x  2  0
a  1, b  4, c  2
b 2  4ac  8
Solutions  2
II.
x 2  3x  10  0
7 x 2  2x  1  0
a  7, b  2, c  1
b 2  4ac  24
Solutions  0
Use the quadratic formula to solve each quadratic equation.
If necessary, round answers to the nearest hundredth.
7.
2x 2  4x  7  0
9.
a  2, b  5, c  3
 5  5 2  4(2)(3)
x
2(2)
 5  1  5 1
x

4
4
a  2, b  4, c  7
 4  4 2  4(2)( 7)
x
2( 2)
 4  72  4  8.49
x

4
4
 1,1.5
1.12,3.12
8.
3x 2  4 x  30
3 x 2  4 x  30  0
a  3, b  4, c  30
 4  4 2  4(3)(30)
x
2(3)
 4   344
x
6
b 2  4ac  0
Solutions : NONE
2x 2  5x  3  0
10.
5 x 2  13x  1  0
a  5, b  13, c  1
 13  13 2  4(5)( 1)
x
2(5)
 13  189  13  13.75
x

10
10
.075,2.68
QUADRATIC FORMULA (CONT)
x
KEY
 b  b 2  4ac
2a
Use the quadratic formula to solve each quadratic equation.
If necessary, round answers to the nearest hundredth.
11.
5 x 2  16 x  84  0
15.
a  5, b  16, c  84
 16  16 2  4(5)( 84)
x
2(5)
 16  1936  16  44
x

10
10
2.8,6
12.
 3x 2  5 x  2  0
.04,14.33
16.
1.5
{. 67,1}
2 x 2  9 x  221  0
17.
13,8.5
x 2  6x  7  0
a  1, b  6, c  7
6  (6) 2  4(1)(7)
x
2(1)
6  8 6  2.83
x

2
2
4.42,1.59
x 2  4 x  10  0
a  1, b  4, c  10
4  (4) 2  4(1)(10)
x
2(1)
4   24
x
2
b 2  4ac  0
Solutions : NONE
a  2, b  9, c  221
9  (9) 2  4(2)( 221)
x
2(2)
9  1849 9  43
x

4
4
14.
4 x 2  12 x  9  0
a  4, b  12, c  9
12  (12) 2  4(4)(9)
x
2(4)
12  0 12  0
x

8
8
a  3, b  5, c  2
 5  5 2  4(3)( 2)
x
2(3)
 5  1  5 1
x

6
6
13.
7 x 2  100 x  4  0
a  7, b  100, c  4
 100  100 2  4(7)( 4)
x
2(7)
 100  10,112  100  100.56
x

14
14
18.
x 2  2x  2  0
a  1, b  2, c  2
 2  2 2  4(1)( 2)
x
2(1)
2 4
x
2
b 2  4ac  0
Solutions : NONE
QUADRATIC FORMULA (CONT)
x
KEY
III.
 b  b 2  4ac
2a
Use the formula ho  16t 2  vt  hi to solve the following.
If necessary, round answers to the nearest hundredth.
19.
Brandy tosses a ball upward with a starting velocity of 10 feet per second. She released
the ball from a height of 3 feet. If the ball is not caught, how long will the ball be in the
air?
0  16t 2  10t  3
 10  292  10  17.09
t

a  16, b  10, c  3
 32
 32

.85,.22
 10  10 2  4(16)(3)
t
t  .85 seconds
2(16)
20.
Jerry threw a ball upward with a starting velocity of 30 feet per second. The ball was 6
feet high when it left his hand. How many seconds (t) did it take for the ball to hit the
ground? 0  16t 2  30t  6
 30  1284  30  35.83
t

a  16, b  30, c  6
 32
 32

2.06,.18
 30  30 2  4(16)(6)
t
t  2.06 seconds
2(16)
21.
A soccer ball bounces straight up into the air off the head of a soccer player at an
altitude of 5 feet with an initial velocity of 30 feet per second. How long will it take for
the ball to hit the ground?
 30  1220  30  34.93
0  16t 2  30t  5
t

 32
 32
a  16, b  30, c  5
2.03,.15
t
22.
 30  30 2  4(16)(5)
2(16)
Benny the frog jumps from one lily pad to the next lily pad. If he jumps with an initial
velocity of 12 feet per second, how long will he take to reach the next lily pad?
0  16t 2  12t  0
a  16, b  12, c  0
 12  12 2  4(16)(0)
t
2(16)
23.
t  2.03 seconds
 12  144  12  12

 32
 32
0,.75
t
t  .75 seconds
Michael shot a 3 point shot. He launched the basketball at 15 feet per second. The ball
left his hand at an altitude of 7 feet. How long would it take the ball to get to the rim
which is 10 feet above the ground?
10  16t 2  15t  7
0  16t 2  15t  3
a  16, b  15, c  3
 15  33  15  5.74

 32
 32
.29,.65
 15  15 2  4(16)( 3)
t
2(16)
t  .65 seconds
t
NOTE: The basketball will leave
Michael’s hand and be at 10 feet
after .29 seconds. It will continue
upward and then come down
and be at the 10 feet height
again (at the rim) after .65
seconds.