Chapter 10
QUADRATIC EQUATIONS,
FUNCTIONS, AND INEQUALITIES
10.1 Solving Quadratic
Equations:
The Square Root
Property of Equality;
Completing the Square
Exercises
2. 281
81 9
x
x
=
=± =±
9, 9xx=− =
8. 2
2
2
260
26
3
3
x
x
x
x
−=
=
=
=±
3, 3xx=− =
14. 2
2
2
10 4 21
411
11
4
11 11
42
p
p
p
i
p
−=
−=
=−
=± − =±
11 11
,
22
ii
pp=− =
16.
()
2
318
318
332
332
x
x
x
x
+=
+=±
+=±
=− ±
332, 332xx=− + =− −
18.
()
2
3 2 108
t
−=
20.
()
()
2
2
21 306
21 24
21 24
2126
2126
12 6
p
p
p
pi
pi
i
p
++=
+=−
+=±−
+=±
=− ±
−±
=
Chapter 10 Quadratic Equations, Functions, and Inequalities
188
24.
2
2
81650
450
nn
n
++=
+=
26.
()
2
2
912427
32 27
32 27
32 33
3233
233
3
xx
x
x
x
x
x
−+=
−=
−=±
−=±
=±
±
=
⎢
⎣
⎡
⎢
⎣
10 10
,
22
ii
tx=− =
30.
2
20
31
31
nn
+= +
32. 2
2
2
1
2
2
2
Emv
Emv
Ev
=
=
34. 22 2
222
ab c
bca
+=
=−
36.
()
2
22
1
16 16 16 64
2
yy yy
⎡⎤
−+⋅− =−+
⎢⎥
⎣⎦
38.
()
2
22
19
33 3
24
xx xx
⎡⎤
−+⋅− =−+
⎢⎥
⎣⎦
22
22
414 42
⎡
⎤⎡⎤
636
66
66
p
p
p
+=±
+=±
=− ±
0, 12pp==−
44.
2
2
24
13
22
13
22
tt
t
t
−=
−=±
=±
Section 10.1 Solving Quadratic Equations: The Square Root Property of Equality; Completing the Square
189
⎡
⎢
⎣
48.
()
2
2
2
4416
8160
40
40
40
4
yyy
yy
y
y
y
y
−=−
−+=
−=
−=±
−=
=
52.
⎣
()
2
2
2
2
2
36 9
3690
3230
230
23
xx
xx
xx
xx
xx
+=−
++=
++=
++=
+=−
625
x
=±
625, 625xx=+ =−
56.
()
2
2
2
2
26100
2350
350
35
tt
tt
tt
tt
+− =
+− =
+−=
+=
58.
()
() ()
2
2
2
22
2
44160
440
40
11
14 1
nn
nn
nn
nn
−+=
−+ =
−+=
⎡
⎤⎡⎤
−+ ⋅− =−+ ⋅−
⎢
⎥⎢⎥
Chapter 10 Quadratic Equations, Functions, and Inequalities
190
60.
⎡
2
2
230
5
23
5
xx
xx
−−=
−=
55
62. 2
2
2
⎡
⎢
⎣
2750
75
20
22
75
0
nn
nn
nn
++=
⎛⎞
++=
⎜⎟
⎝⎠
++=
2
64. 2
2
2
2
315
32 15
2
315
3
25
rrr
rr
rr
rr
−=−
−=−
⎛⎞
−=−
⎜⎟
⎝⎠
−=−
64. (continued)
2
144
39
r
⎛⎞
−=−
⎜⎟
⎝⎠
()
2
2
22
2151
16
16
16
xx
x
x
x
⎢
⎥⎢⎥
⎣
⎦⎣⎦
++=+
+=
+=±
=− ±
122
122
x
x
−=±
=±
122, 122xx=+ =−
70. The coefficient of the linear term n is 5.−
()
15
5
22
−=−
Section 10.1 Solving Quadratic Equations: The Square Root Property of Equality; Completing the Square
191
72.
()
2
2
2
2
43 32
16 24 9 32
16 24 23
3
16 23
t
tt
tt
tt
−=
−+=
−=
⎛⎞
−=
⎜⎟
74.
2
2
16 16 4 22
12 38
xx x
xx
++=−
+=−
76. 2
2
124
2
48
48 4 3
x
x
x
=
=
=± =±
43, 43xx==−
80. Area of a circle is 2.
A
r
π
=
Let r be the radius of the wading pool.
()
()
2
2
3.14 2 785
2250
r
r
+=
+=
()
2
10 25 3000 25
53025
53025
xx
x
x
++= +
+=
+=±
b.
() ()
() ( )
2
5 100 0.5 100
5 100 0.5 10,000
500 5000
5500
R=+
=+
=+
=
No, since the revenue is $5500, which is not
equal to
()
2 $1500 , or $3000.
Chapter 10 Quadratic Equations, Functions, and Inequalities
192
84. b.
()
2
16 2 10.3 3400
3389.76 0
h=− ⋅ +
=− ≠
2
0 16 3400
t
=− +
86. Let x be the speed of the officer walking
west. Distance = rate × time. Since time is
given in seconds, the rate of 30 ft per min
must be converted to ft per sec.
ft ft 1 ft
30 30
min 60 sec 2 sec
==
86. (continued)
2
11481
2161616
rr
++=+
problem.
3ft 3 ft 360 ft 90 ft/min
1
2sec 2 2 min
min
60
⋅
== =
90 30 120+=
The security officer walking south walks at
a rate of 120 ft/min, and the security officer
walking west walks at a rate of 90 ft/min.
Mindstretchers
1. No. Taking the cube root of each side gives only
one of the solutions, namely 1. There are also two
complex solutions (containing i). The solutions
are found by factoring and completing the square.
3
Section 10.2 Solving Quadratic Equations: The Quadratic Formula
193
2. Answers may vary.
3. 2
2
60
6
xxc
xxc
++=
+=−
c. For solutions that are not real, the right
side of the equation, when completing
the square, must be negative.
The solutions are not real when 9.c>
10.2 Solving Quadratic
Equations:
The Quadratic Formula
Exercises
2. The discriminant is the radicand in the
quadratic formula.
4. If a quadratic equation has two real
solutions, its discriminant is positive.
8. 2840xx+−=
() () ()( )
2
1, 8, 4
88414
abc
x
===−
−± − −
=
10. 2
2
25
520
1, 5, 2
xx
xx
ab c
=−
+−=
===−
() () ()( )
()
2
55412
21
5258
2
533
2
533 533
,
22
x
xx
−± − −
=
−± +
=
−±
=
−+ −−
==
12. 2
2
2116
250
1, 2, 5
xx
xx
ab c
++=
++=
== =
() () ()()
()
2
22415
21
x
−± −
=
Chapter 10 Quadratic Equations, Functions, and Inequalities
194
14. 2
2
2101
21010
2, 10, 1
xx
xx
ab c
−=
−−=
==− =−
16. 2
2
8573
8230
8, 2, 3
nnn
nn
ab c
+=+
−−=
==−=−
18. 2
41070
4, 10, 7
xx
ab c
−+=
==− =
()() ()()
()
2
10 10 4 4 7
24
x
−− ± − −
=
20. 2
2
31485
3650
3, 6, 5
xxx
xx
abc
−+ =−
−++=
=− = =
33
22. 22
2
13 8 2 4 2 3
9610
9, 6, 1
pp pp
pp
abc
+−= +−
++=
===
()
2
2
2
991
93
39
390
xx
xx
xx
⎛⎞
+=−
⎜⎟
⎝⎠
+=−
++=
1, 3, 9abc===
Section 10.2 Solving Quadratic Equations: The Quadratic Formula
195
26.
()
2
2
2
111
0
8412
111
24 24 0
8412
3620
xx
xx
xx
−−=
⎛⎞
−− =
⎜⎟
⎝⎠
−−=
3, 6, 2ab c==−=−
28.
()
()
2
2
2
0.5 0.3 0.2 0
10 0.5 0.3 0.2 10 0
5320
xx
xx
xx
+−=
+−=
+−=
5, 3, 2abc===−
30.
()
()
2
2
2
0.02 0.16 0.34 0
100 0.02 0.16 0.34 100 0
xx
xx
++=
++=
30. (continued)
()
16 16
4
16 4
4
44
x
i
i
−±−
=
−±
=
−±
()
2
2210
2
21 10
2
±
=
±
=
12 180
18
12 6 5
18
−±
=
−±
=
Chapter 10 Quadratic Equations, Functions, and Inequalities
196
36. 2
0.6 4.9 3.3 0
0.6, 4.9, 3.3
xx
ab c
−−=
==−=−
38. 2
2.04 0.45 0.017 0
2.04, 0.45, 0.017
xx
abc
++=
===
40. 2
730
7, 1, 3
xx
ab c
−+=
==−=
() ()()
2
241473
184
83
bac−=−−
=−
=−
Two complex solutions (containing i)
One real solution
44. 2
2
625
6520
xx
xx
=−
+−=
46. 2
8710
8, 7, 1
xx
ab c
−−=
==−=−
() ()()
056
56
=−
=−
Two complex solutions (containing i)
52. 22
2
23278
4850
xx x x
xx
−−=−−
−−=
4, 8, 5ab c==−=−
() () ()()
()
2
88445
24
82 82
x
−− ± − − −
=
54. 2
0.8 3.7 0.5 0
0.8, 3.7, 0.5
xx
ab c
−−=
==−=−
Section 10.2 Solving Quadratic Equations: The Quadratic Formula
197
56. 2
2
0.05 48 100 9000
0.05 48 9100 0
xx
xx
−+−=
−+−=
0.05, 48, 9100abc=− = =−
58.
()( )
2
2
11 13 2 315
143 35 2 315
2 35 172 0
xx
xx
xx
++=
++=
+−=
2, 35, 172ab c== =−
60. Distance = rate × time. Speed is in mph,
time must be in hrs. 30 min = ½ hr.
60.
()
22
2
11
42 315
22
mm
⎛⎞⎛ ⎞
++=
⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
1
22
21 441 197,568
1
21 198, 009
⎜⎟
⎝⎠
−± +
=
=− ±
0.288
2.067 1.547433
0.288
2.067 1.547433 11
x
−±
=−
−−
=≈
Chapter 10 Quadratic Equations, Functions, and Inequalities
198
Mindstretchers
1. a. 2
2330xx+−=
2, 3, 3ab c== =−
() ()
()( )
()
2
33423
22
x
−± − −
=
43 23 3
3;
442
xx
−
==− ==
b.
260xix−+=
1, , 6abic==−=
2. It will have two real solutions if
2
.
4
b
ca
<
For example, consider the equation
230,xx+−=
()
2
1
3 and 3 .
41
c=− − <
3. Answers may vary.
10.3 More on Quadratic
Equations
Exercises
2. An equation is quadratic in form if it can be
6.
2
22
2
2
76
3
76
3
xx
xx
xx
−=
⎛⎞⎛⎞
−=
⎜⎟⎜⎟
⎝⎠⎝⎠
2
?
?
33
33
37 6
22
21 9
36
24
⎛⎞
−−
⎜⎟
⎝⎠
⎛⎞⎛⎞
−− =−
⎜⎟⎜⎟
⎝⎠⎝⎠
⎛⎞
+=
⎜⎟
⎝⎠
Solutions:
2,3
3
−
Section 10.3 More on Quadratic Equations
199
8.
31
10
62
tt
−−=
+−
Check 0.t=
?
?
31
10
06 02
31
10
62
−−=
+−
+−=
10. 42
280aa+−=
Let 2.ua=
() ()
2
22
280
aa
+−=
10. (continued)
Check 2 .ai=
16 8 8 0
00
−−=
=
Check 2.a=
() ()
?
42
22280
+−=
12. 42
9526yy−+=
2
Let .uy=
() ()
2
22
9526
yy
−+=
Chapter 10 Quadratic Equations, Functions, and Inequalities
200
12. (continued)
3
Check 1.y=
() ()
() ()
?
42
?
?
91 51 2 6
91 51 2 6
9526
66
−+=
−+=
−+=
=
Check 1.y=−
2
Check .
3
i
y=
42
?
22
9526
ii
⎛⎞ ⎛⎞
−+=
⎜⎟ ⎜⎟
2
Check .
3
i
y=−
42
?
22
9526
33
ii
⎛⎞⎛⎞
−−−+=
⎜⎟⎜⎟
⎝⎠⎝⎠
14. 20 0xx−−=
54
uu
==−
ux=
54
25 0
xx
xx
==−
=≥
4− is not a solution.
Check 25.x=
?
?
25 25 20 0
−−=
() ()
()( )
2
14 14
2
320
320
120
xx
uu
uu
−+=
−+=
−−=
Check 1.x=
() ()
?
12 14
?
13120
1320
−+=
−+=
Section 10.3 More on Quadratic Equations
201
18. 23 13 12 0yy−−=
13
Let .uy=
()()
2
13 13
12 0
yy
−−=
Check 27.y=−
() ()
()
?
23 13
?
27 27 12 0
93120
00
−−−−=
−− − =
=
20.
()()
2
24 230pp++ ++=
Let 2.up=+
()()
2
2
24 230
430
pp
uu
++ ++=
++=
20. (continued)
Check 3.p=−
()()
?
2
?
2
32 4 32 30
−+ + −+ + =
()()
()()
()()
2
2
32 632 8
32 63280
680
240
xx
xx
uu
uu
−− −=−
−− −+=
−+=
−−=
20 40
uu
−= −=
Check 2.x=
()
()
()
()
?
?
32 2 6 32 2 8
62 662 8
−− −=−
−− −=−
Chapter 10 Quadratic Equations, Functions, and Inequalities
202
24. 36
30 60
xx
xx
==
−= −=
()()
2
360
9180
xx
xx
−−=
−+=
30. 25
52
25
52
250
520
pp
p
p
p
p
==
=
=
−=
−=
()()
2
52250
10 29 10 0
pp
pp
−−=
−+=
36. 66
60 60
xi x i
xi xi
==−
−= +=
()()
()
2
2
2
660
60
36 0
xixi
xi
x
−+=
−=
+=
38.
2
22
2
2
53
2
53
2
52 3
xx
xx
xx
xx
+=
⎛⎞⎛⎞
+=
⎜⎟⎜⎟
⎝⎠⎝⎠
+=
() ()
?
52 2 32 339
11
10 2 3 4 33
12 12
+= − + =
+=⋅ =
=
Solutions: 1
3, 2
−
Check 9.b=
() ()
?
?
?
959240
953240
915240
00
+−=
+⋅− =
+−=
=
Solution: 9
Section 10.3 More on Quadratic Equations
203
42. 77
70 70
xx
xx
=− =−
+= +=
()()
()
2
2
770
70
14 49 0
xx
x
xx
++=
+=
++=
46. Distance = rate time
distance
time , distance = 9 miles
rate
×
=
Let rate of of the currentr=.
()()()()
()()
()() ()()
99
0.5
88
98 8 98 8
88
0.5 8 8
98 98 0.58 8
rr
rr rr
rr
rr
rr rr
=−
+−
+− +−
=
+−
−+−
−= +− + −
46. (continued)
() () ( )( )
()
2
18 18 4 0.5 32
20.5
18 324 64
1
x
−± − −
=
−± +
=
()( )( ) ()( )( )
()( )( )
()()()
2
2
2
11 1
45 30
1453014530
45
14530
30
30 45 30 45
30 1350 30 45
1350 60 45
tt
tt tt
tt
tt
tttt
tttt
tt t
+=
+
++
++
+
=
++ =+
++=+
+=+