Chapter 5 Practice Quiz Solutions
For each equation, find all rational solutions. If the problem has no solution, so state.
1.
x2 + x – 30 = 0
(x + 6)(x – 5) = 0
2.
Solution set = { –6, 5 }
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Solution set = { 4, 9 }
x2 – 13x + 36 = 0
(x – 9)(x – 4) = 0
3.
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x2 + 25 – 10x = 0
x2 – 10x + 52 = 0
(x – 5)2
( binomial2 )
=0
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Solution set = { 5 }
4. (y – 3)(y + 7) = 0
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Solution set = { –7, 3 }
5.
x3 + x2 + 25x + 25 = 0
( factor by grouping )
x2(x + 1) + 25(x + 1) = 0
(x2 + 25)(x + 1) = 0
(x2 + 52)(x + 1) = 0
6.
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Solution set = { –1 }
t7 = 4t5
t7 – 4t5 = 0
t5(t2 – 4) = 0
( difference of squares )
t5(t2 – 22) = 0
t5(t + 2)(t – 2) = 0
7.
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Solution set = { –2, 0, 2 }
a3 + 3a2 = 4a
a3 + 3a2 – 4a = 0
a(a2 + 3a – 4) = 0
a(a + 4)(a – 1) = 0
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Solution set = { –4, 0, 1 }
8.
28x – 48 + 10x2 = 0
10x2 + 28x – 48 = 0
2(5x2 + 14x – 24) = 0
2(5x – 6)(x + 4) = 0
9.
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4t2 – 25 = 0
Solution set = { –4, 6/5 }
( difference of squares )
(2t)2 – 52 = 0
(2t + 5)(2t – 5) = 0
10.
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Solution set = { –5/2, 5/2 }
x (x – 1) = 6
x2 – x = 6
x2 – x – 6 = 0
(x + 2)(x – 3) = 0
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11. (m – 9)(m – 2)(m + 3) = 0
Solution set = { –2, 3 }
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Solution set = { –3, 2, 9 }
12. Factor completely: 3r3 – 3 = 3(r3 – 1)
= 3(r3 – 13)
( difference of cubes )
= 3(r – 1)(r2 + r + 1)
13.
45r2 + 60r + 20 = 0
5(9r2 + 12r + 4) = 0
( binomial2 )
5((3r)2 + 2(3r)(2) + 22 ) = 0
5(3r + 2)2 = 0
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Solution set = { –2/3 }
3x4 – 48 = 0
14.
3(x4 – 16) = 0
( difference of squares )
3((x2)2 – 42) = 0
3(x2 + 4)(x2 – 4) = 0
( difference of squares again)
3(x2 + 22)(x2 – 22) = 0
3(x2 + 22)(x + 2)(x – 2) = 0
15. Factor completely:
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Solution set = { –2, 2 }
ab + ac – db – dc
( factor by grouping )
= a(b + c) – d(b + c)
= (a – d)(b + c)
16.
Factor completely:
x4 + 2x3 – 3x – 6
( factor by grouping )
= x3 (x+ 2) – 3(x + 2)
= (x3 – 3)(x + 2)
x2 + 3x + 6 = 0
17.
18.
Since two integers whose product is 6 can't add up to 3,
this equation has no solution.
4x (x – 1) = 15
4x2 – 4x = 15
4x2 – 4x – 15 = 0
(2x – 5)(2x + 3) = 0
19.
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Solution set = { –3/2, 5/2 }
2t2 – 5 = –3t
2t2 + 3t – 5 = 0
(2t + 5)(t – 1) = 0
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Solution set = { –5/2, 1 }
20. Factor completely:
m2 – mn – 20n2
= (m + 4n)(m – 5n)
21. Find the x-intercepts for the graph of the parabola y = x2 – 13x + 22.
Recall that the x-intercepts occur when y = 0:
x2 – 13x + 22 = 0
(x – 2)(x – 11) = 0
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x-intercepts at (2, 0) and (11, 0).
22. The length of a rectangle is 6 more than the width. The area of the rectangle is 40 m2.
Find the rectangle's length and width.
W (W + 6) = 40
W2 + 6W = 40
W2 + 6W – 40 = 0
(W + 10)(W – 4) = 0
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Width = 4m, Length = 10m
23. The number of possible handshakes H within a group of n people is given by:
H = (n2 – n) / 2
At a meeting, everyone shook hands with everyone else. If there were 45 handshakes,
how many people attended the meeting?
45 = (n2 – n) / 2
90 = n2 – n
0 = n2 – n – 90
0 = (n + 9)(n – 10)
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10 people