Post-Algebra II Summer Review
Name___________________________________
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Show your work in the space provided. Circle your final answer.
Period____
I. Simplify each expression.
1)
(n + 2)(n + 5)
n-1
×
6n(n + 2)
n+5
n 2 - 49 n 2 + n - 72
3)
×
10n + 70 n 2 + 2n - 63
2)
(m + 5)(m + 9) 1
×
4 m 2 (m + 4 ) m + 9
a-9
a 2 - 3a - 10
4) 2
×
a + 4a + 4 a 2 - 8a + 15
Worksheet by Kuta Software LLC
-1-
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5)
3n 2 + 21n + 36
5n + 7
× 2
30n + 42
3n + 3n - 18
6)
7 x 2 + 32 x - 60 24 x 3 + 32 x 2
×
24 x 3 + 32 x 2
7 x - 10
7)
2 x + 8 x 2 + 13 x + 36
¸
6 x - 60
x - 10
8)
2 p 2 + 20 p p 2 + 16 p + 60
¸
2 p 2 - 20 p
3 p + 18
Worksheet by Kuta Software LLC
-2-
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9)
8r - 8 5r 2 + 7r - 24
¸
20r - 20
50r - 80
10)
9k 2 + 90k
56k - 64
¸
2
7k - 8
49k - 112k + 64
II. Simplify each complex fraction.
x-4
11)
25
4
x-4
5
12)
x
4
x2
Worksheet by Kuta Software LLC
-3-
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u
4
4 u-1
13)
16
+
14)
u2
2
u
+
5
4
u
x+3
4
16
3x - 5
15)
8
x+3
x+3
16
1
-
16)
-
m
m2
m-2
25
m-2
m-2
m
Worksheet by Kuta Software LLC
-4-
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III. Simplify all radicals.
17)
12b 4
19) 6 288n 5
18)
486b 2
20) -8 324n 4
Worksheet by Kuta Software LLC
-5-
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21) 3 18 x 3 y
23)
3
-1250 x
22) 3 90 x 5 y
24)
5
288b 7
Worksheet by Kuta Software LLC
-6-
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IV. Rationalize each denominator. Simplify all radicals.
25)
9 3
8
27) -
7
7 10
26)
5 2
10 7
28)
4 18
6 42
Worksheet by Kuta Software LLC
-7-
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29)
2
7
30)
6
8 10
V. Solve each equation. Remember to check for extraneous solutions.
31) 6 =
m
32) 2 +
x=9
Worksheet by Kuta Software LLC
-8-
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33)
7m = 3m + 4
34)
4x + 1 = 5x
35)
6-x=x
36)
4x = x
Worksheet by Kuta Software LLC
-9-
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37)
37 - 2n = n - 1
39) 2 -
1 - 8v = 3 - 2v
38)
4m - 32 = m - 7
40)
3n + 4 = -1 - 2n + 1
Worksheet by Kuta Software LLC
-10-
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VI. Solve each equation by factoring.
41) 8 x = -15 - x 2
42) n 2 = -6 - 5n
43) 0 = 20m - 48 - 2m 2
44) 3n 2 = 3n + 90
Worksheet by Kuta Software LLC
-11-
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45) 6r 2 - r - 7 = 8
46) 8 x 2 - 35 x + 5 = 6 x
47) 21k 2 - 30 = 7k 2 + 32k
48) -2 - 4m = -6m 2
Worksheet by Kuta Software LLC
-12-
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VII. Solve each equation by taking square roots.
49) r 2 = 49
50) r 2 = 36
51) -9r 2 = -9
52) m 2 - 6 = 10
Worksheet by Kuta Software LLC
-13-
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53) -r 2 = -11
54) k 2 + 2 = 67
55) 64 x 2 - 5 = 59
56) 10v 2 - 10 = 240
Worksheet by Kuta Software LLC
-14-
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VIII. Solve each equation with the quadratic formula.
57) 2k 2 + 8k - 24 = 0
58) 6m 2 + 8m - 40 = 0
59) 20k 2 - 29 = -5k + 12k 2 - 5
60) -3 - 7b = -12b 2
Worksheet by Kuta Software LLC
-15-
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Answers to Show your work in the space provided. Circle your final answer.
n-1
6n
n+4
5)
6( n - 2 )
1)
4
r+3
u 3 - u 2 + 16u
13)
64u - 64
-24m - 2
16) 3
m + 4m - m 2 - 4
20) -144n 2
5
24) 2b 9b 2
9)
2 21
21
32) {49}
36) {0, 4}
28)
40) No solution.
44) {6, -5}
{ }
1
3
52) {4, -4}
56) {5, -5}
48) - , 1
59)
{
m+5
4m 2 (m + 4)
6) x + 6
2)
3)
n-8
4)
a-9
(a + 2)(a - 3)
10)
9k(k + 10)
8
3
4u
14)
8 + 5u
17) 2b 2 3
10
1
3
7)
8)
3( x + 9 )
p - 10
2
x - 8 x + 16
5x
11)
12)
100
4
3
2
3 x + 13 x - 67 x - 237
15)
387 x - 3 x 3 - 13 x 2 - 595
18) 9b 6
19) 72n 2 2n
21) 9 x 2 xy
22) 9 x 2 10 xy
9 6
4
14
29)
7
33) {1}
37) {6}
14
14
15
30)
40
34) {1}
38) {9}
25)
26)
41) {-5, -3}
31) {36}
35) {2}
{{ } }
{{ } }
{
55) {1, -1}
{ }
53) { 11 , - 11 }
57) {2, -6}
54)
}
10
10
1
,5
8
50) {6, -6}
46)
-5 + 793 -5 - 793
,
16
16
27) -
1
9
43) 4, 6
5
47) - , 3
7
51) 1, -1
42) {-2, -3}
3 5
45) - ,
2 3
49) {7, -7}
3
23) -5 10 x
{ }
65 , - 65 }
10
58) 2, 3
7 + 193 7 - 193
60)
,
24
24
39)
{ }
{
}
Worksheet by Kuta Software LLC
-16-
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Examples and YouTube Links for Each Section
I. Simplify each expression.
x 2  3 x  10
x7
 2
2
x  8 x  15 x  4 x  4
( x  5)( x  2)
x7


( x  5)( x  3) ( x  2)( x  2)


( x  5) ( x  2)
( x  5) ( x  3)

( x  7)
( x  2) ( x  2)
( x  7)
( x  3)( x  2)
x 2  3 x  10 x 2  4 x  4

x7
x 2  8 x  15
2
x  3 x  10
x7
 2
2
x  8 x  15 x  4 x  4
( x  5)( x  2)
x7


( x  5)( x  3) ( x  2)( x  2)


( x  5) ( x  2)
( x  5) ( x  3)

( x  7)
( x  2) ( x  2)
( x  7)
( x  3)( x  2)
Multiplying and dividing rational expressions: https://www.youtube.com/watch?v=sfhypVsrr3c
II. Simplify each complex fraction.
1
x
x
1
x
x
1 x2

x
x

1 x2

x x
1  x2
 x 2
1 x
x
x
1  x2


x 1  x2
x
1  x2


x 1  x2
1  x2

1  x2
Adding and subtracting rational expressions: https://www.youtube.com/watch?v=Nx9xoplP2Dg
III. Simplify all radicals.
3
3 32 x5 y 3
 3 16  2  x  x  x  y  y
2
2
1
2
54 x 7
 3 27  2  x3  x 3  x1
1
 3 33  2  x3  x3  x1
 3  4  x  x  y 2 x1 y1
 3 x  x 3 2x
 12 x 2 y 2 xy
 3x 2 3 2 x
Simplifying square roots:
https://www.youtube.com/watch?v=fN4g3X1LTa0
Simplifying roots with larger indices:
https://www.youtube.com/watch?v=XUw4gOxSLO8
IV. Rationalize each denominator. Simplify all radicals.
2
8 10

2
10

8 10 10

2 10
8 10 10

20
8 100
45
8 10
2 5

80
2 5

80


5
40
Rationalizing the denominator: https://www.youtube.com/watch?v=3yDYCVh6orA
V. Solve each equation. Remember to check for extraneous solutions.
8m  2 m  6
3 x  8

33 x  83
x 5
 x
2
8m
 
2
2m  6

2
8m  2 m  6
8m  2 m  2 m  2 m  6
6m  6
6m 6

6
6
m 1
Check: Let m  1.
 52
x  25
Check: Let x  25.
3  25  8
35  8
88
8 1  2 1  6
 x  25
8 8
m  1
7  4 x  32  x
7  7  4 x  32  x  7
4 x  32  x  7

4 x  32

2
  x  7   FOIL
2
4 x  32  x 2  14 x  49
0  x 2  14 x  49  4 x  32
0  x 2  18 x  81  Factor
0  ( x  9)( x  9)
x 9  0
x 99  09
x9
Check: Let x  9.
7  4  9  32  9
7 4 9
729
99
x  9
Solving radical equations: https://www.youtube.com/watch?v=e26DDsnK5Nk
https://www.youtube.com/watch?v=MkDZgd1SOfI
https://www.youtube.com/watch?v=8FZUQgEBhz8
VI. Solve each equation by factoring.
x 2  17 x  30
x 2  17 x  30  30  30
x 2  17 x  30  0
 x  ____  x  ____   0
Factors of  30 whose sum is  17 ?
 x  15 x  2   0
x  15  0 or x  2  0
x  15  15  0  15 or x  2  2  0  2
x  15 or
x2
______________________________________________________________________________
6 x2  2  4 x
6 x2  4 x  2  4 x  4 x
6 x2  4 x  2  0
2(3 x 2  2 x  1)  0
2  0, so 3 x 2  2 x  1  0
Solve: 3 x 2  2 x  1  0
a  c  (3)(1)  3
Factors of  3 whose sum is  2?
Re-write  2 x using  3 and  1; factor.
3 x 2  3 x  1x  1  0
3
3x 
1x  1  0
x 2



GCF
GCF
3 x( x  1)  1( x  1)  0
( x  1)(3 x  1)  0
x  1  0 or 3 x  1  0
x  1 or
3 x  1
3 x 1

3
3
1
x
3
Solving by factoring:
https://www.youtube.com/watch?v=s4r0qlnyGcE
https://www.youtube.com/watch?v=4OD8Dc7Ey7A
https://www.youtube.com/watch?v=fSlYrV75kUY
VII. Solve each equation by taking square roots.
3x 2  51  57
3x 2  51  51  57  51
3x 2  108
3x 2 108

3
3
2
x  36
x   36
x  6
Square Root Property:
http://mathbyfives.com/mathbyfives/Algebra/Entries/2010/8/1_Square_Root_Property.html
VIII. Solve each equation with the quadratic formula.
Quadratic formula:
b  b 2  4ac
2a
3 x 2  3  12 x
3x 2  12 x  3  12 x  12 x
3 x 2  12 x  3  0
3( x 2  4 x  1)  0
3  0, so x 2  4 x  1  0
a  1, b  4, c  1
x
4  42  4(1)(1)
2(1)
4  20
2
4  4  5
x
2
4  2 5
x
2
4 2 5
x

2
2
x  2  5
x
Solve by using the quadratic formula: https://www.youtube.com/watch?v=S0XXBaECixo
Remembering the quadratic formula: https://www.youtube.com/watch?v=-OK3r9FiZHs