Document 1804401

Chapter 4,
continued
5. x 2 1 22x 1 c
78. F;
3
3x 2 2y 5 10 l y 5 }2 x 2 5 l y-intercept 5 (0, 25)
25 2 1
26
0 2 (24)
3
So m 5 2}2 , b 5 25.
c 5 1}
5 112 5 121
22
22 2
3
m5}5}
5 2}2
4
x 2 1 22x 1 121 5 (x 1 11)(x 1 11) 5 (x 1 11)2
6. x 2 2 9x 1 c
29 2
c 5 1}
5}
2 2
4
Lesson 4.7
81
x 2 2 9x 1 }
5 1 x 2 }2 21 x 2 }2 2 5 1 x 2 }2 2
4
81
Investigating Algebra Activity 4.7 (p. 283)
9
9
9 2
7. x 2 1 6x 1 4 5 0
1.
x 2 1 6x 5 24
Completing the Square
x 1 6x 1 9 5 24 1 9
2
Number of 1-tiles
needed to complete
the square
Expression
written as a
square
x 2 1 2x 1 ?
1
x 2 1 2x 1 1
5 (x 1 1)2
x 2 1 4x 1 ?
4
x 2 1 4x 1 4
5 (x 1 2)2
9
x 1 6x 1 9
5 (x 1 3)2
16
x 2 1 8x 1 16
5 (x 1 4)2
25
x 1 10x 1 25
5 (x 1 5)2
Expression
(x 1 3)2 5 5
x 2 1 8x 1 ?
2. a. The value of d is one half the value of b.
8.
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
x 2 2 10x 1 8 5 0
x 2 2 10x 1 25 5 28 1 25
(x 2 5)2 5 17
}
x 2 5 5 6Ï17
}
x 5 5 6 Ï 17
}
}
The solutions are 5 1 Ï 17 and 5 2 Ï 17 .
9. 2n 2 2 4n 2 14 5 0
n 2 2 2n 2 7 5 0
n 2 2 2n 5 7
c. You can multiply the value of b by one half and then
n 2 2 2n 1 1 5 7 1 1
square the result.
(n 2 1)2 5 8
4.7 Guided Practice (pp. 285–287)
}
n 2 1 5 6Ï 8
1. x 1 6x 1 9 5 36
2
}
n 5 1 6 Ï8
(x 1 3)2 5 36
}
n 5 1 6 2Ï 2
x 1 3 5 66
}
}
The solutions are 1 1 2Ï 2 and 1 2 2Ï2 .
x 5 23 6 6
The solutions are 23 1 6 5 3 and 23 2 6 5 29.
10. 3x 2 1 12x 2 18 5 0
x 2 1 4x 2 6 5 0
2. x 2 2 10x 1 25 5 1
x 2 1 4x 5 6
(x 2 5)2 5 1
x 1 4x 1 4 5 6 1 4
2
x 2 5 5 61
(x 1 2)2 5 10
x5561
}
x 1 2 5 6Ï 10
The solution are 5 1 1 5 6 and 5 2 1 5 4.
}
x 5 22 6 Ï 10
3. x 2 2 24x 1 144 5 100
}
}
The solutions are 22 1 Ï 10 and 22 2 Ï10 .
(x 2 12)2 5 100
x 2 12 5 610
11.
6x(x 1 8) 5 12
6x 2 1 48x 5 12
x 5 12 6 10
The solutions are 12 1 10 5 22 and 12 2 10 5 2.
4. x 2 1 14x 1 c
x 2 1 8x 5 2
x 2 1 8x 1 16 5 2 1 16
(x 1 4) 2 5 18
c 5 1}
5 72 5 49
22
2
x 1 14x 1 49 5 (x 1 7)(x 1 7) 5 (x 1 7)
2
}
x 2 2 10 5 28
b. The value of c is the square of the value of d.
14
}
The solutions are 23 1 Ï 5 and 23 2 Ï5 .
2
x 2 1 10x 1 ?
}
x 5 23 6 Ï 5
2
x 2 1 6x 1 ?
}
x 1 3 5 6Ï 5
2
}
x 1 4 5 63Ï 2
}
x 5 24 6 3Ï 2
}
}
The solutions are 24 1 3Ï 2 and 24 2 3Ï2 .
Algebra 2
Worked-Out Solution Key
209
Chapter 4,
4p(p 2 2) 5 100
4. x 2 1 10x 1 25 5 64
4 p 2 8p 5 100
(x 1 5)2 5 64
2
p 2 2 2p 5 25
x 1 5 5 68
p 2 2p 1 1 5 25 1 1
x 5 25 6 8
2
( p 2 1) 5 26
The solutions are 25 1 8 5 3 and 25 2 8 5 213.
2
}
p 2 1 5 6Ï26
5. n 2 1 16n 1 64 5 36
}
p 5 1 6 Ï26
}
(n 1 8)2 5 36
}
The solutions are 1 1 Ï26 and 1 2 Ï26 .
13.
n 1 8 5 66
n 5 28 6 6
y 5 x 2 2 8x 1 17
The solutions are 28 1 6 5 22 and 28 2 6 5 214.
y 1 16 5 (x2 2 8x 1 16) 1 17
6. m 2 2 2m 1 1 5 144
y 1 16 5 (x 2 4)2 1 17
(m 2 1)2 5 144
y 5 (x 2 4)2 1 1
The vertex form of the function is y 5 (x 2 4)2 1 1.
The vertex is (4, 1).
14.
7. x2 2 22x 1 121 5 13
(x 2 11)2 5 13
y 1 9 5 (x 1 3)2 1 3
The vertex form of the function is y 5 (x 1 3) 2 6.
The vertex is (23, 26).
2
f (x) 5 x 2 2 4x 2 4
f (x) 1 4 5 (x 2 2 4x 1 4) 2 4
f (x) 1 4 5 (x 2 2)2 2 4
f (x) 5 (x 2 2)2 2 8
The vertex form of the function is f (x) 5 (x 2 2)2 2 8.
The vertex is (2, 28).
y 5 216(t 2 2 5t) 1 2
1
4
2
y 2 100 5 2161 t 2 }2 2 1 2
5
2
y 5 2161 t 2 }2 2 1 102
5 2
The vertex is 1 }2 , 102 2, so the maximum height of the
5
baseball is 102 feet.
4.7 Exercises (pp. 288–291)
Skill Practice
1. A binomial is the sum of two monomials and a trinomial
is the sum of three monomials.
2. For an expression of the form x 2 1 bx, you complete the
square by first finding half of b and squaring the result.
Then you add the result to the expression.
3. x 2 1 4x 1 4 5 9
(x 1 2) 5 9
2
x 1 2 5 63
x 5 22 6 3
The solutions are 22 1 3 5 1 and 22 2 3 5 25.
210
Algebra 2
Worked-Out Solution Key
}
}
The solutions are 11 1 Ï13 and 11 2 Ï 13 .
8. x 2 2 18x 1 81 5 5
(x 2 9)2 5 5
}
x 2 9 5 6Ï 5
}
x 5 9 6 Ï5
}
}
The solutions are 9 1 Ï 5 and 9 2 Ï 5 .
9. t 2 1 8t 1 16 5 45
}
t 1 4 5 63Ï 5
25
25
y 5 (216) } 5 216 t 2 2 5t 1 } 1 2
}
x 5 11 6 Ï 13
(t 1 4)2 5 45
y 5 216t 2 1 80t 1 2
142
}
x 2 11 5 6Ï 13
y 5 (x 1 3)2 2 6
16.
m 5 1 6 12
The solutions are 1 1 12 5 13 and 1 2 12 5 211.
y 5 x 2 1 6x 1 3
y 1 9 5 (x2 1 6x 1 9) 1 3
15.
m 2 1 5 612
}
t 5 24 6 3Ï 5
}
}
The solutions are 24 1 3Ï 5 and 24 2 3Ï5 .
10. 4u 2 1 4u 1 1 5 75
(2u 1 1)2 5 75
}
2u 1 1 5 65Ï 3
}
2u 5 21 6 5Ï 3
1
}
5Ï3
u 5 2}2 6 }
2
1
}
5Ï 3
}
5Ï3
1
and 2}2 2 }
.
The solutions are 2}2 1 }
2
2
11. 9x 2 2 12x 1 4 5 23
(3x 2 2)2 5 23
}
3x 2 2 5 6Ï23
}
3x 5 2 6 Ï 23
}
3x 5 2 6 i Ï3
}
2
i Ï3
2
i Ï3
x 5 }3 6 }
3
}
2
}
i Ï3
and }3 2 }
.
The solutions are }3 1 }
3
3
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
12.
continued
Chapter 4,
continued
12. A;
x 2 1 4x 5 10
22.
x 2 4x 1 4 5 21
x 1 4x 1 4 5 10 1 4
2
(x 2 2) 5 21
2
(x 1 2)2 5 14
2
}
x 2 2 5 6Ï21
x 5 2 6 Ï21
}
x 2 1 8x 5 21
23.
x 2 1 8x 1 16 5 21 1 16
c 5 1 }2 2 5 32 5 9
6 2
(x 1 4)2 5 15
x 1 6x 1 9 5 (x 1 3)(x 1 3) 5 (x 1 3)
2
2
}
24. x 2 1 6x 2 3 5 0
x 2 1 12x 1 36 5 (x 1 6)(x 1 6) 5 (x 1 6)2
x 2 1 6x 5 3
15. x 2 2 24x 1 c
x 2 1 6x 1 9 5 3 1 9
224 2
c5 }
5 (212)2 5 144
2
2
(x 1 3)2 5 12
}
x 5 23 6 Ï 12
16. x 2 2 30x 1 c
}
x 5 23 6 2Ï 3
230 2
c5 }
5 (215)2 5 225
2
2
}
}
The solutions are 23 1 2Ï 3 and 23 2 2Ï3 .
x 2 2 30x 1 225 5 (x 2 15)(x 2 15) 5 (x 2 15)2
25. x 2 1 12x 1 18 5 0
x 2 1 12x 5 218
17. x 2 2 2x 1 c
x 2 1 12x 1 36 5 218 1 36
22 2
c5 }
5 (21)2 5 1
2
1 2
(x 1 6)2 5 18
}
x 1 6 5 6Ï 18
x 2 2 2x 1 1 5 (x 2 1)(x 2 1) 5 (x 2 1)2
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
}
x 1 3 5 6Ï 12
x 2 2 24x 1 144 5 (x 2 12)(x 2 12) 5 (x 2 12)2
}
x 5 26 6 Ï 18
18. x 2 1 50x 1 c
}
x 5 26 6 3Ï 2
1 2
50 2
c5 }
5 252 5 625
2
}
}
The solutions are 26 1 3Ï 2 and 26 2 3Ï2 .
x 2 1 50x 1 625 5 (x 1 25)(x 1 25) 5 (x 1 25)2
26. x 2 2 18x 1 86 5 0
x 2 2 18x 5 286
19. x 2 1 7x 1 c
x 2 18x 1 81 5 286 1 81
2
c 5 1 }2 2 5 }
4
49
(x 2 9)2 5 25
1
21
2 1
2
49
7
7
7 2
5 x 1 }2 x 1 }2 5 x 1 }2
x 2 1 7x 1 }
4
}
x 2 9 5 6Ï25
}
x 5 9 6 Ï 25
20. x 2 13x 1 c
2
}
x 5 9 6 i Ï5
}
c 5 1}
5}
2 2
4
169
}
The solutions are 9 1 i Ï 5 and 9 2 i Ï5 .
5 1x 2 }
x2}
5 x2}
x 2 2 13x 1 }
4
2 21
22 1
22
169
13
13 2
13
27. x 2 2 2x 1 25 5 0
x 2 2 2x 5 225
x 2 2 2x 1 1 5 225 1 1
21. x 2 2 x 1 c
(x 2 1)2 5 224
c 5 1 2}2 2 5 }4
1
}
x 21 5 6Ï 224
x 2 2 x 1 }4 5 1 x 2 }2 21 x 2 }2 2 5 1 x 2 }2 2
1
}
The solutions are 24 1 Ï 15 and 24 2 Ï15 .
12 2
1 2
}
x 5 24 6 Ï 15
c 5 1}
5 62 5 36
22
213 2
}
x 1 4 5 6Ï 15
14. x 2 1 12x 1 c
7 2
}
The solutions are 22 1 Ï 14 and 22 2 Ï 14 .
13. x 2 1 6x 1 c
1
}
x 5 22 6 Ï 14
x526i
1
}
x 1 2 5 6Ï 14
}
1
1
1 2
}
x 5 1 6Ï 224
}
x 5 1 6 2i Ï 6
}
}
The solutions are 1 1 2i Ï6 and 1 2 2i Ï6 .
Algebra 2
Worked-Out Solution Key
211
Chapter 4,
continued
2k 2 1 16k 5 212
33. 6r 2 1 6r 1 12 5 0
r2 1 r 1 2 5 0
r 2 1 r 5 22
k 2 1 8k 5 26
k 1 8k 1 16 5 26 1 16
2
(k 1 4)2 5 10
1
k 1 4 5 6Ï10
1 r 1 }12 2
2
}
k 5 24 6 Ï10
}
}
The solutions are 24 1 Ï 10 and 24 2 Ï 10 .
1
3x 1 42x 5 224
x 2 1 14x 5 28
x 1 14x 1 49 5 28 1 49
}
}
}
The solutions are 27 1 Ï 41 and 27 2 Ï 41 .
x 2 10x 2 3 5 0
2
x2 2 10x 5 3
x 2 10x 1 25 5 3 1 25
2
Î
1
i Ï7
1
i Ï7
}
7
}
}
}
i Ï7
1
34. C;
}
x 1 5 5 6Ï 12
}
x 5 25 6 Ï 12
}
x 2 5 5 6Ï28
}
x 5 25 6 2Ï 3
}
x 5 5 6 Ï28
35. Area of rectangle 5 *w 5 50
}
x 5 5 6 2Ï7
}
(x 1 10)(x) 5 50
}
The solutions are 5 1 2Ï 7 and 5 2 2Ï7 .
x 2 1 10x 5 50
31. 3s 2 1 6s 1 9 5 0
x 2 1 10x 1 25 5 50 1 25
s 2 1 2s 1 3 5 0
s 2 1 2s 5 23
2
4s 1 2s 1 1 5 23 1 1
(s 1 1)2 5 22
(x 1 5)2 5 75
}
x 1 5 5 6Ï 75
}
x 5 25 6 Ï 75
}
x 5 25 6 5Ï 3
}
}
The value of x is 25 1 5Ï3 .
}
s 5 21 6 Ï22
36. Area of parallelogram 5 bh 5 48
}
s 5 21 6 i Ï 2
}
}
The solutions are 21 1 i Ï2 and 21 2 i Ï2 .
32. 7t 2 1 28t 1 56 5 0
t 2 1 4t 1 8 5 0
t 2 1 4t 5 28
2
t 1 4t 1 4 5 28 1 4
(t 1 2)2 5 24
(x 1 6)(x) 5 48
x 2 1 6x 5 48
x 2 1 6x 1 9 5 48 1 9
(x 1 3)2 5 57
}
x 1 3 5 6Ï 57
}
x 5 23 6 Ï 57
}
The value of x is 23 1 Ï57 .
}
t 1 2 5 6Ï24
}
t 5 22 6 Ï24
t 5 22 6 2i
The solutions are 22 1 2i and 22 2 2i.
Algebra 2
Worked-Out Solution Key
1
x 2 1 10x 1 8 5 25
x 2 1 10x 5 213
2
x 1 10x 1 25 5 213 1 25
(x 1 5)2 5 12
30. 4x 2 40x 2 12 5 0
2
212
7
and 2}2 2 }
.
The solutions are 2}2 1 }
2
2
}
x 5 27 6 Ï41
s 1 1 5 6Ï22
}
r 5 2}2 6 }
2
x 1 7 5 6Ï41
(x 2 5)2 5 28
Î
r 5 2}2 6 2}4
2
(x 1 7)2 5 41
7
5 2}4
r 1 }2 5 6 2}4
2
29.
1
r 2 1 r 1 }4 5 22 1 }4
}
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
28.
Chapter 4,
continued
1
37. Area of triangle 5 } bh 5 40
2
1
} (x)(x 1 4) 5 40
2
1
} x(x 1 4) 5 40
2
1
} x 2 1 2x 5 40
2
42.
y 1 4 5 1 x 2 2 4x 1 4 2 2 1
y 1 4 5 (x 2 2)2 2 1
y 5 (x 2 2)2 2 5
The vertex form of the function is y 5 (x 2 2)2 2 5.
The vertex is (2, 25).
x 2 1 4x 5 80
x 2 1 4x 1 4 5 80 1 4
(x 1 2)2 5 84
43.
y 1 36 5 (x 1 6) 2 1 37
}
y 5 (x 1 6) 2 1 1
x 5 22 6 Ï 84
}
The vertex form of the function is y 5 (x 1 6)2 1 1.
The vertex is (26, 1).
x 5 22 6 2Ï21
}
The value of x is 22 1 2Ï 21 .
1
38. Area of trapezoid 5 } (b1 1 b2)h 5 20
2
1
} (x 1 9 1 3x 2 1)(x) 5 20
2
1
} x(4x 1 8) 5 20
2
44.
y 1 100 5 (x 1 10)2 1 90
y 5 (x 1 10) 2 2 10
The vertex form of the function is y 5 (x 1 10)2 2 10. The
vertex is (210, 210).
x 2 1 2x 5 10
x 1 2x 1 1 5 10 1 1
2
(x 1 1)2 5 11
45.
9
}
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
h 5 216t 2 1 89.6t
h 5 216 (t 2 2 5.6t)
h 1 (216)(7.84) 5 216 (t 2 2 5.6t 1 7.84)
h 2 125.44 5 216(t 2 2.8)2
h 5 216(t 2 2.8)2 1 125.44
The vertex of the function’s graph is (2.8, 125.44).
This means that at 2.8 seconds, the water will reach its
maximum height of 125.44 feet.
y 5 0.0085x 2 2 1.5x 1 120
y 5 0.0085(x 2 2 176.47x) 1 120
y 1 (0.0085)(7785.42)
5 0.0085(x 2 2 176.47x 1 7785.42) 1 120
y 1 66.18 5 0.0085(x 2 88.24)2 1 120
y 5 0.0085(x 2 88.24)2 1 53.82
The vertex of the function’s graph is (88.24, 53.82). This
means that when you walk about 88.24 meters per minute,
your rate of energy use will reach a minimum of 53.82
calories per minute.
y 5 x 2 8x 1 19
2
9
f (x) 1 }4 5 1 x 2 }2 2 1 4
9
x 5 21 6 Ï 11
The value of x is 21 1 Ï 11 .
41.
f (x) 5 x 2 2 3x 1 4
f (x) 1 }4 5 1 x 2 2 3x 1 }4 2 1 4
}
x 1 1 5 6Ï 11
}
y 5 x 2 1 20x 1 90
y 1 100 5 (x 2 1 20x 1 100) 1 90
2x 2 1 4x 5 20
40.
y 5 x 2 1 12x 1 37
y 1 36 5 (x 2 1 12x 1 36) 1 37
}
x 1 2 5 6Ï 84
39.
y 5 x 2 2 4x 2 1
3 2
f (x) 5 1 x 2 }2 2 1 }4
3 2
7
The vertex form of the function is f (x) 5 1 x 2 }2 2 1 }4 .
3 2
7
The vertex is 1 }2, }4 2.
3 7
46.
g(x) 5 x 2 1 7x 1 2
g(x) 1 }
5 1 x 2 1 7x 1 }
12
4
42
49
49
5 1 x 1 }2 2 1 2
g(x) 1 }
4
7 2
49
g(x) 5 1 x 1 }2 2 2 }
4
7 2
41
.
The vertex form of the function is g(x) 5 1 x 1 }2 2 2 }
4
7 2
41
.
The vertex is 1 2}2, 2}
42
7
47.
41
y 5 2x 2 1 24x 1 25
y 5 2(x 2 1 12x) 1 25
y 1 (2)(36) 5 2(x 2 1 12x 1 36) 1 25
y 1 16 5 (x 2 8x 1 16) 1 19
2
y 1 72 5 2(x 1 6)2 1 25
y 1 16 5 (x 2 4) 1 19
2
y 5 2(x 1 6)2 2 47
y 5 (x 2 4) 1 3
2
The vertex form of the function is y 5 2(x 1 6)2 2 47.
The vertex form of the function is y 5 (x 2 4) 1 3.
The vertex is (4, 3).
2
The vertex is (26, 247).
Algebra 2
Worked-Out Solution Key
213
Chapter 4,
continued
y 5 5x 2 1 10x 1 7
53. x 2 1 3x 1 14 5 0
y 5 51 x 2 1 2x 2 1 7
y 1 (5)(1) 5
51 x 2
x 2 1 3x 5 214
1 2x 1 1 2 1 7
9
y 1 5 5 5(x 1 1) 1 7
1 x 1 }32 2
2
y 5 5(x 1 1)2 1 2
The vertex form of the function is y 5 5(x 1 1)2 1 2.
The vertex is (21, 2).
q 2 1 2q 5 31
q 2 1 2q 1 1 5 31 1 1
(q 1 1)2 5 32
last step.
}
q 1 1 5 6Ï32
}
Ï 12 5 Ï 4 + Ï 3 5 2Ï 3
}
q 5 21 6 4Ï 2
x 1 10x 1 13 5 0
2
}
}
The solutions are 21 1 4Ï 2 and 21 2 4Ï2 .
x 2 1 10x 5 213
x 2 1 10x 1 25 5 213 1 25
3x 2 1 x 5 2x 2 6
55.
3x 2 2 x 5 26
(x 1 5)2 5 12
}
x 1 5 5 6Ï12
1
x 2 2 }3 x 5 22
}
x 5 25 6 Ï12
}
x 5 25 6 2Ï3
1
1
1
5 22 1 }
x2 2 }3 x 1 }
36
36
1 x 2 }16 2
2
51. The method of completing the square was done incorrectly.
Because 4(9), or 36, is added to the left side, it must also
be added to the right side.
71
5 2}
36
Î
}
1
71
x 2 }6 5 6 2}
36
4x 1 24x 2 11 5 0
2
}
i Ï 71
1
4(x 2 1 6x) 5 11
x 5 }6 6 }
6
4(x 2 1 6x 1 9) 5 11 1 36
}
i Ï 71
1
1
}
i Ï71
and }6 2 }
.
The solutions are }6 1 }
6
6
4(x 1 3)2 5 47
47
(x 1 3)2 5 }
4
56. 0.1x 2 2 x 1 9 5 0.2x
}
Ï
0.1x 2 2 1.2x 5 29
47
x1356 }
4
}
Ï47
x 5 23 6 }
2
x 2 2 12x 5 290
x 2 12x 1 36 5 290 1 36
2
(x 2 6)2 5 254
}
52. x 1 9x 1 20 5 0
2
x 2 6 5 6Ï254
}
x 2 1 9x 5 220
x 5 6 6 3i Ï6
81
81
x 2 1 9x 1 }
5 220 1 }
4
4
1
Î
}
1
x 1 }2 5 6 }4
9
1
x 5 2}2 6 }2
The solutions are 24 and 25.
214
}
i Ï47
5q 2 1 10q 5 155
}
9
3
7q 2 1 10q 5 2q 2 1 155
54.
50. The error was made when simplifying Ï 12 in the
5 }4
}
i Ï 47
3
The vertex form of the function is y 5 2(x 2 7)2 1 1.
The vertex is (7, 1).
2
}
i Ï47
and 2}2 2 }
.
The solutions are 2}2 1 }
2
2
y 5 2(x 2 7)2 1 1
1 x 1 }92 2
47
3
y 1 98 5 2(x 2 7)2 1 99
}
}
x 5 2}2 6 }
2
y 1 (2)(49) 5 2(x 2 2 14x 1 49) 1 99
}
Î
3
y 5 2(x 2 2 14x) 1 99
}
47
5 2}
4
x 1 }2 5 6 2}
4
y 5 2x 2 2 28x 1 99
49.
9
x 2 1 3x 1 }4 5 214 1 }4
2
Algebra 2
Worked-Out Solution Key
}
}
The solutions are 6 1 3i Ï6 and 6 2 3i Ï 6 .
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
48.
Chapter 4,
continued
57. 0.4v 2 1 0.7v 5 0.3v 2 2
Problem Solving
0.4v2 1 0.4v 5 22
h 5 216t 2 1 32t 1 6
62.
v 1 v 5 25
2
h 5 2161 t 2 2 2t 2 1 6
h 1 (216)(1) 5 2161 t 2 2 2t 1 1 2 1 6
1
1
v 2 1 v 1 }4 5 25 1 }4
1
2
h 2 16 5 216(t 2 1)2 1 6
19
1 2
5 2}
2
4
v1}
h 5 216(t 2 1)2 1 22
Î
}
1
The maximum height of the baton is 22 feet.
19
v 1 }2 5 6 2}
4
}
i Ï 19
1
h 5 216t 2 1 48t 1 4
63.
}
1
i Ï 19
v 5 2}2 6 }
2
1
h 5 216(t 2 2 3t) 1 4
}
i Ï19
h 1 (216)1 }4 2 5 2161 t 2 2 3t 1 }4 2 1 4
h 2 36 5 2161 t 2 }2 2 1 4
3 2
58. Sample answer: x 2 1 6x 5 1
59. a.
y
y
h 5 2161 t 2 }2 2 1 40
3 2
The maximum height of the volleyball is 40 feet.
1
1
y 5 (x 1 1)2
1
21
x
y 5 (70 2 x)(50 1 x)
64.
x
y 5 x2 1 2x
y 5 3500 1 70x 2 50x 2 x 2
y 5 (x 1 2)2
y 5 3500 1 20x 2 x 2
y 5 x2 1 4x
y 5 2x 2 1 20x 1 3500
y
6
9
9
and 2}2 2 }
.
The solutions are 2}2 1 }
2
2
y 5 2(x 2 2 20x) 1 3500
y 5 (x 2 3)2
y 1 (21)(100) 5 2(x 2 2 20x 1 100) 1 3500
y 2 100 5 2(x 2 10)2 1 3500
21
y 5 2(x 2 10)2 1 3600
x
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
y 5 x 2 2 6x
The shop can maximize weekly revenue by decreasing the
price per skateboard by $10. With this decrease in price,
the weekly revenue will be $3600.
b. When you complete the square, there is a vertical
1 2
b 2
translation of the graph of y 5 x 2 1 bx, }2 units up.
60. 1 real solution: k 5 0
2 real solutions: k > 0
2 imaginary solutions: k < 0
61.
x 2 1 bx 1 c 5 0
x 2 1 bx 5 2c
x 2 1 bx 1 1 }2 2 5 2c 1 1 }2 2
b 2
1 x 1 }b2 2
2
b
b 2
b2
5}
2c
4
Î
}
b2
2c
x 1 }2 5 6 }
4
b
Î
6Î
}
b2
2c
x 5 2}2 6 }
4
b
x 5 2}2
}
b 2 2 4c
4
}
}
b
Ïb 2 2 4c
x 5 2}2 6 }
2
65.
y 5 (200 1 10x)(40 2 x)
y 5 8000 2 200x 1 400x 2 10x 2
y 5 8000 1 200x 2 10x 2
y 5 210x 2 1 200x 1 8000
y 5 210(x 2 2 20x) 1 8000
y 1 (210)(100) 5 210(x 2 2 20x 1 100) 1 8000
y 2 1000 5 210(x 2 10)2 1 8000
y 5 210(x 2 10)2 1 9000
The store can maximize monthly revenue by increasing the
price per video game system by 10x 5 10(10) 5 $100. With
this increase in price, the monthly revenue will be $9000.
66. a.
y 5 20.0110x 2 1 1.23x 1 5.50
y 5 20.0110(x2 2 111.82x) 1 5.50
y 1 (20.0110)(3125.93)
5 20.0110(x 2 2 111.82x)
1 3125.93 1 5.50
y 2 34.39 5 20.0110(x 2 55.91)2 1 5.50
y 5 20.0110(x 2 55.91)2 1 39.89
}
2b 6 Ïb 2 2 4c
x 5 }}
2
Algebra 2
Worked-Out Solution Key
215
Chapter 4,
continued
Volume of clay:
b.
x
0
10
20
y
5.50
16.71
25.71
x
70
80
90
y
37.71
33.51
30
40
50
Vclay 5 Voutside 2 Vinside
60
200 5 9:(x 1 6x 1 9) 2 9:(9 2 x)
32.51 37.11 39.51 39.71
100
27.11 18.51
110
120
7.71
25.29
200 5 9: 1 x 2 1 6x 1 9 2 9 1 x 2
200 5 9: 1 x 2 1 7x 2
200
9:
}F
5 x 2 1 7x
y
y 5 20.0110(x 2 55.91)2 1 39.89
40
200
9:
49
4
1
200
9:
49
4
1
32
Height (ft)
49
4
}F
1 } 5 x 2 1 7x 1 }
}
Ï
7
49
200
1}
5x
2}2 6 Î}
4
9:
200
49
7
6 }
F
1}
5 x 1 }2
4
9:
16
}
8
0
0
24
48
72
96
120
The maximum height of the softball is 39.89 feet.
The ball travels a distance of 116.13 feet.
67. a. Area of cutting section 5 *w
1500 5 x(120 2 2x)
b.
x(120 2 2x) 5 1500
120x 2 2x 2 5 1500
22x 2 1 120x 5 1500
x 2 2 60x 5 2750
2
x 2 60x 1 900 5 2750 1 900
(x 2 30)2 5 150
}
x 2 30 5 6Ï150
23.5 6 4.4 ø x
x
Horizontal position (ft)
Reject the negative value, 23.5 2 4.4, or 27.9.
The pencil holder should have a thickness of about
0.9 centimeter.
Mixed Review for TAKS
69. A;
If the quadrilateral is reflected in the line y 5 3, the image
of point N will be the same as the original point N. The
image of point N will be in Quadrant I.
70. H;
Gallons in small pool
Hours to fill small pool
}
120
1.5
}
Length 5 30 1 5Ï6 ø 42.25
The dimensions of the eating section are 35.51 feet by
42.25 feet.
68. Volume of cylinder 5 :r 2h
Outside cylinder: r 5 x 1 3
h59
Voutside 5 :(x 1 3)2(9) 5 9: 1 x 2 1 6x 1 9 2
Inside cylinder: r 5 3
h592x
Vinside 5 :(3) (9 2 x) 5 9:(9 2 x)
2
600
h
}5}
}
You must reject 30 2 5Ï 6 , or about 17.75. This
value of x gives a width of about 84.5 feet. A width of
84.5 feet is not possible because the side of the school
is 70 feet.
}
c. Width 5 120 2 21 30 1 5Ï 6 2 ø 35.51
Gallons in large pool
Hours to fill large pool
}} 5 }}
x 5 30 6 5Ï6
120h 5 900
h 5 7.5
It will take 7.5 hours to fill the 600 gallon pool.
Quiz 4.5–4.7 (p. 291)
1. 4x 2 5 64
2. 3( p 2 1)2 5 15
x 2 5 16
( p 2 1)2 5 5
}
x 5 6Ï 16
x 5 64
4. 22z 2 5 424
1
(m 1 5)2 5 }2
Î
z 2 5 2212
}
1
m 1 5 5 6 }2
}
Ï
}
1
m 5 25 6 }2
}
Ï2
5. s 2 1 12 5 9
}
s 5 6Ï23
}
s 5 6 i Ï3
Algebra 2
Worked-Out Solution Key
}
p 5 1 6 Ï5
3. 16(m 1 5)2 5 8
s 2 5 23
}
p 2 1 5 6Ï5
m 5 25 6 }
2
216
2
7 2
2
}F
1}5 x1}
24
2
z 5 6Ï2212
}
z 5 6 i Ï 212
}
z 5 6 2i Ï53
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
c.
Chapter 4,
continued
6. 7x 2 2 4 5 26
7x 5 22
y 1 49 5 (x 2 1 14x 1 49) 1 45
y 1 49 5 (x 1 7)2 1 45
2
x 2 5 2}7
y 5 x 2 1 14x 1 45
14.
2
y 5 (x 1 7)2 2 4
Î
}
2
x 5 6 2}7
The vertex form of the function is y 5 (x 1 7)2 2 4.
The vertex is (27, 24).
}
Ï
2
x 5 6 i }7
f(x) 5 x 2 2 10x 1 17
15.
}
f (x) 1 25 5 (x 2 2 10x 1 25) 1 17
Ï 14
x 5 6 i}
7
f(x) 1 25 5 (x 2 5)2 1 17
f(x) 5 (x 2 5)2 2 8
7. (5 2 3i) 1 (22 1 5i) 5 [5 1 (22)] 1 (23 1 5)i
The vertex form of the function is f(x) 5 (x 2 5)2 2 8.
The vertex is (5, 28).
5 3 1 2i
8. (22 1 9i) 2 (7 1 8i) 5 (22 2 7) 1 (9 2 8)i
16.
5 29 1 i
g(x) 1 1 5 (x 2 2 2x 1 1) 2 7
9. 3i(7 2 9i) 5 21i 2 27i 2
g(x) 1 1 5 (x 2 1)2 2 7
5 21i 2 27(21)
g(x) 5 (x 2 1)2 2 8
5 27 1 21i
10. (8 2 3i)(26 2 10i) 5 248 2 80i 1 18i 1 30i
The vertex form of the function is g(x) 5 (x 2 1)2 2 8.
The vertex is (1, 28).
2
5 248 2 62i 1 30(21)
17.
5 278 2 62i
1
1
1 2
3
The vertex form of the function is y 5 1 x 1 }2 2 1 }4.
1 2
244 2 24i
Copyright © by McDougal Littell, a division of Houghton Mifflin Company.
1 2
y 5 1 x 1 }2 2 1 }4
224i 1 44(21)
5 }}
36 2 121(21)
3
The vertex is 1 2}2, }4 2.
1 3
5}
157
24
2}
i
5 2}
157
157
18.
3 2 2i
3 2 2i
28 2 5i
12. } 5 } + }
28 1 5i 28 2 5i
28 1 5i
y 5 x 2 1 9x 1 19
5 1 x 2 1 9x 1 }
1 19
y1}
4
42
81
224 2 15i 1 16i 1 10i 2
64 1 40i 2 40i 2 25i
81
5 1 x 1 }2 2 1 19
y1}
4
81
5 }}
2
224 1 i 1 10(21)
9 2
y 5 1 x 1 }2 2 2 }4
9 2
5 }}
64 2 25(21)
5
The vertex form of the function is y 5 1 x 1 }2 2 2 }4 .
234 1 i
9 2
5}
89
5
The vertex is 1 2}2, 2}4 2.
9
1
1}
i
5 2}
89
89
13.
1
y 1 }4 5 1 x 1 }2 2 1 1
224i 1 44i 2
36 2 66i 1 66i 2 121i
5 }}2
34
y 5 x2 1 x 1 1
y 1 }4 5 1 x 2 1 x 1 }4 2 1 1
4i
26 1 11i
4i
11. } 5 } + }
26 2 11i 26 1 11i
26 2 11i
44
g(x) 5 x 2 2 2x 2 7
y 5 x 2 2 4x 1 9
19.
5
h 5 216t 2 1 h0
0 5 216t 2 1 45
y 1 4 5 (x 2 2 4x 1 4) 1 9
245 5 216t 2
y 1 4 5 (x 2 2)2 1 9
y 5 (x 2 2)2 1 5
The vertex form of the function is y 5 (x 2 2) 1 5.
The vertex is (2, 5).
2
45
16
} 5 t2
Î45
}
6 }
5t
16
61.7 ø t
Reject the negative solution, 21.7, because time must be
positive. The ball is in the air for about 1.7 seconds.
Algebra 2
Worked-Out Solution Key
217