Chapter 10
QUADRATIC EQUATIONS,
FUNCTIONS, AND INEQUALITIES
Equations:
The Square Root
Property of Equality;
Exercises
4. 263
n
=
6. 2
64 0
p
+=
2
26
x
=
10. 2
a
=
46, 46aa=− =
12. 2
2
51954
535
y
y
+=
=
2
2
411
11
p
p
−=
=−
11 11
,
22
ii
pp=− =
()
318
332
332
x
x
x
+=±
+=±
=− ±
()
3 2 108
3
t
−=±
()
()
21 24
p
+=±−
()
()
2
36
10 25
36
10 25
y
y
−=
−=±
188
24.
()
81650
450
nn
n
++=
+=±
()
2
32 27
32 27
3233
x
x
x
−=
−=±
=±
()()
2
2
16 9 49
16 40
16 4
t
t
−=−
=−
()
31 20
31 25
3125
125
n
n
n
+=±
+=±
=− ±
−±
32. 2
2
2
2
Emv
=
22
bca
=± −
()
2
⎢⎥
⎣⎦
()
2
24
⎢⎥
⎣⎦
40.
⎢
⎥⎢⎥
42.
()
2
22
12 0
636
pp
p
+=
+=±
() ()
22
2
2
2
11
12 1
22
181
444
22
13
tt
tt
⎡
⎤⎡⎤
−+ ⋅− = + ⋅−
⎢
⎥⎢⎥
⎣
⎦⎣⎦
−+ = +
Section 10.1 Solving Quadratic Equations: The Square Root Property of Equality; Completing the Square
() ()
()
22
33
xx
x
−+=
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
−=
33, 33xx=+ =−
48.
()
2
4416
40
yyy
y
−=−
−=
50. 2
2
2
235106
261035
825
pp p
ppp
pp
++=−
++=−
+=−
()
49
p
+=±−
()
()
2
22
3230
23
12
xx
xx
x
++=
+=−
+=±−
54.
()
() ()
()
2
12 16
22
62025
625
xx
xx
x
x
−+=
−=−
⎢
⎥⎢⎥
−=± =±
=±
56.
()
2
26100
3353
22
9209
3444
242
329
tt
tt
tt
+− =
++ ⋅ =+ ⋅
⎢⎥ ⎢⎥
⎣⎦ ⎣⎦
++= +
58.
()
() ()
2
44160
40
22
11515
242
nn
nn
i
n
−+=
−+=
⎢
⎥⎢⎥
⎣
⎦⎣⎦
−=±− =±
190
60.
2
22
5
23
5
55
xx
xx
x
−−=
−=
=±
2
2
75
20
22
nn
n
⎛⎞
++=
⎜⎟
⎝⎠
+=± =±
64. 2
2
315
315
3
25
3
rrr
rr
rr
−=−
−=−
⎜⎟
⎝⎠
−=−
2
144
39
144
r
r
⎛⎞
−=−
⎜⎟
⎝⎠
−=±−
66.
()
2
35
25
16
xx x
xx
x
+−=
+=
+=±
68.
()
() ()
()
25 14
22
xxx
⎢
⎥⎢⎥
⎣
⎦⎣⎦
70. The coefficient of the linear term n is 5.−
()
2
15
5
22
−=−
Section 10.1 Solving Quadratic Equations: The Square Root Property of Equality; Completing the Square
2
2
16 24 23
3
16 23
2
16 23 9
4
4
342
tt
tt
t
t
−=
⎛⎞
−=
⎜⎟
⎝⎠
−=+
⎜⎟
⎝⎠
=±
74.
()
2
2
16 16 4 22
62
xx x
x
++=−
+=±−
2
48
x
=
() ()
22
2
9 16 2500
25 2500
xx
x
+=
=
() ()
The screen is 30 in. by 40 in.
()
()
2
2250
r
+=
2 5 10 13.8, 2 5 10 17.8rr=− + ≈ =− − ≈−
A negative answer makes no sense in this
()
2
22
10 3000
xx
+=
()
53025
x
+=±
50, 60xx==−
A negative answer makes no sense in this
() ( )
500 5000
=+
84. Note that the negative answers make no sense
in this problem, so we disregard them.
The supplies will be 1700 ft above the ground
about 10.3 seconds later.
192
3389.76 0
=− ≠
2
0 16 3400
3400 10 34 5 34 14.6
t
=− +
86. Let x be the speed of the officer walking
west. Distance = rate × time. Since time is
()
2
2
22
22
10 10 25
2
1
100 100 625
2
1
100 100 625
4
100 100 100 25 625
rr
rr
rrr
rrr
⎛⎞
++=
⎜⎟
⎜⎟
⎝⎠
⎝⎠
⎛⎞
++=
⎜⎟
⎝⎠
⎛⎞
+++=
⎜⎟
⎝⎠
+++=
2161616
rr
++=+
3ft 3 ft 360 ft 90 ft/min
1
2sec 2 2 min
⋅
== =
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.
()
()
10
1
x
x
−=
=
2
2
22
2
10
1
11
111
22
xx
xx
xx
++=
+=−
⎡
⎤⎡⎤
++ ⋅ =−+ ⋅
⎢
⎥⎢⎥
⎣
⎦⎣⎦
2
11
1
xx
++ =−+
193
3. 2
2
22
60
6
xxc
xxc
++=
+=−
It has a single solution when 9.c=
The solutions are not real when 9.c>
Equations:
The Quadratic Formula
Exercises
6. 27120xx−+=
() () ()()
()
21
74948
2
71
x−− ± − −
=
±−
=
±
() () ()( )
()
()
2
88414
21
2
x
−± − −
=
=
10. 2
25
xx
=−
() () ()( )
()
21
2
533 533
,
22
x
xx
=
=
−+ −−
==
()
2
24
2
212
i
i
=
−±
=
−±
=
194
2
21010
xx
−−=
()() ()()
()
()
22
10 100 8
4
4
x−− ± − − −
=
±+
=
=
16. 2
() () ()()
()
22483
28
2496
16
2100
n−− ± − − −
=
±+
=
±
16 2 16 4
18. 2
()() ()()
()
()
24
10 100 112
8
10 12
8
x
=
±−
=
±−
=
20. 2
2
3650
xx
−++=
() () ( )()
()
()
()
23
63660
6
23 2 6
23
x
=−
−± +
=−
−±
=−
() () ()()
()
29
63636
18
60
p
=
−± −
=
−±
24.
1
93
93
⎝⎠
1, 3, 9abc===
() () ()()
()
2
33419
21
3936
2
x
−± −
=
−± −
=
195
26.
8412
() () ()()
()
()
()
2
66432
23
63624
6
23 15
23
x
−− ± − − −
=
±+
±
=
()
() () ()( )
()
3940
37
−± +
−±
=
()
2
216340
xx
++=
2, 16, 34ab c== =
() () ()()
()
2
16 16 4 2 34
22
16 256 272
4
x−± −
=
−± −
=
()
16 16
4
4, 4
x
xixi
−±−
=
=
=− + =− −
32.
()()
241
xx
+−=
() () ()()
()
21
2436
2
21 10
x−− ± − − −
=
±+
=
±
()()
2
96126
xx x
++=−
() () ()( )
()
()
()
29
18
62 5
63
25 25
,
33
x
xx
=
=
−±
=
−+ −−
==
Chapter 10 Quadratic Equations, Functions, and Inequalities
196
0.6, 4.9, 3.3
ab c
==−=−
=
40. 2
730
7, 1, 3
xx
ab c
−+=
==−=
83
=−
Two complex solutions (containing i)
42. 28160
1, 8, 16
xx
ab c
−+=
==−=
8, 7, 1
ab c
==−=−
2
4850
xx
−−=
() () ()()
()
24
86480
8
8 144
x
=
±+
=
±
=
0.131, 4.756
xx
±
≈− ≈
197
()
2
20.05
0.10
58.
()( )
2
2
11 13 2 315
143 35 2 315
2 35 172 0
xx
xx
xx
++=
++=
+−=
2, 35, 172ab c== =−
()
2
21 441 197,568
1
21 198,009
−± +
=
=− ±
0.288
x
−
The function is defined for the years 2003
through 2009, or 0x= through 6.x=
Therefore, disregard 11.x=
198
2, 3, 3ab c== =−
() ()
()( )
()
2
33423
3324
4
327
4
−± − −
−±+
=
−±
=
43 23 3
3;
442
xx
==− ==
() () ()()
()
21
124
2
25
x
i
i
=
±−−
=
±−
=
2. It will have two real solutions if
.
4
ca
<
For example, consider the equation
2
(containing i) if
.
4
b
ca
>
For example, consider the equation
3. Answers may vary.
Equations
Exercises
rewritten in the form 20,au bu c++= where
0a≠ and u represents an algebraic expression.
4. To find a quadratic equation whose solutions are
6.
()()
2
3
3760
32 30
xx
xx
xx
−−=
+−=
320
32
2
3
x
x
x
+=
=−
=−
30
3
x
x
−=
=
2
?
33
37 6
22
⎝⎠
⎛⎞⎛⎞
−− =−
⎜⎟⎜⎟
⎝⎠⎝⎠
Check 3.x=
?
2
76
333
−=
199
()() ()()()
62
62 621
62
tt
tt tt
tt
+−
+− − =+−
⎢⎥
+−
⎣⎦
()()()
()
2
2
36 6 412
212 412
tt tt
ttt
−−−= + −
−=+−
0t= 20
2
t
t
+=
=−
?
10
62
336
0
666
00
+−=
+−=
=
?
?
31
10
44
314
0
+−=
+−=
Let 2.ua=
() ()
()()
420
uu
+−=
40 20
42
uu
uu
+= −=
=− =
2
ua
=
Check 2 .ai=
()
?
00
=
Check 2 .ai=−
()
42
?
16 2 4 8 0
16 8 8 0
ii
+−=
−−=
() ()
()
?
?
16 2 4 8 0
4480
+−=
+−=
()()
()
?
?
16 2 4 8 0
+−=
2
Let .uy=
() ()
()( )
2
9526
9540
19 4 0
yy
uu
uu
−+=
−−=
−+=
10 9 4 0
uu
−= + =
200
2
9
2
3
uy
i
y
=
=±
Check 1.y=
Check 1.y=−
2
Check .
y=
9526
33
9
66
−+=
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
=
=
2
Check .
3
i
y=−
42
?
81 9 9
16 20 18 6
⎝⎠⎝⎠
++=
14. 20 0xx−−=
()()
()( )
50 40
54
uu
uu
−= +=
==−
ux=
?
25 5 20 0
−− =
16. 12 14
() ()
()( )
10 20
uu
−= − =
() ()
116
xx
==
Check 1.
x=
() ()
?
12 14
13120
−+=
Check 16.
() ()
?
12 14
16 3 16 2 0
−+=
201
13
Let .uy=
()()
()()
30 40
uu
+= −=
()
()
()
()
34
27 64
yy
yy
=− =
=− =
() ()
()
?
23 13
?
27 27 12 0
93120
00
−−−−=
−− − =
=
Check 64.y=
?
00
=
()()
()()
2
2
24 230
430
310
pp
uu
uu
++ ++=
++=
++=
Check 5.p=−
()()
() ()
?
2
?
2
52 4 52 30
34330
−+ + −+ + =
−+−+=
Check 3.p=−
1430
00
−+=
=
()
()()
()()
()()
2
2
32 632 8
680
240
xx
uu
uu
−− −=−
−+=
−−=
20 40
24
uu
uu
−= −=
==
32ux=−
() ()
324 3216
xx
−= −=
()
()
?
?
62 662 8
464 8
−− −=−
−=−
()
()
?
?
18 2 6 18 2 8
16 6 16 8
−− −=−
−=−
202
()()
()( )
1100
nn
−+=
28. 15
450
41
aa
a
a
=− =−
+=
=−
()()
30. 25
pp
==
()()
2
10 29 10 0
pp
−+=
()()
()
2
2
80
16 64 0
y
yy
+=
++=
34. 55
xx
==−
()()
()
2
2
2
50
50
x
x
−=
−=
()()
()
2
2
2
60
36 0
xi
x
−=
+=
()()
210 30
xx
−= + =
1
Check . Check 3.
13
xx==−
Solutions: 1
3, 2
−
() ()
()()
2
5240
5240
830
bb
uu
uu
+−=
+−=
+−=
80 30
83
uu
uu
+= −=
=− =
83
09
bb
bb
=− =
≥=
8
− is not a solution.
?
953240
915240
00
+⋅− =
+−=
=
Solution: 9
Section 10.3 More on Quadratic Equations
203
()
2
14 49 0
xx
++=
()
()( )
2
1 10, 000 11, 290
r
+=
() () ()( )
2 2 4 1 0.129
24.516
0.06254, 2.06254
r
−± − −
−±
46. Distance = rate time
×
Let rate of of the currentr=.
()
99
=
0.5, 18, 32abc===−
18 324 64
1
−± +
=
this problem. The rate of the current is
approximately 1.7 mph.
enced
tt
()( )( ) ()( )( )
()( )( )
()()()
()()
11 1
45
14530
45 30 0
tt
tt
tt
++
+
−+=
The negative answer makes no sense in this
204
Mindstretchers
1. a.
22
22
44
2
2
2
aa
b b ac b b ac
a
bb
aa
+
−+ − −− −
=
−−
==
The sum is .
b
a
− The sum is the ratio
()()
()
()
()
2
2
22
2
2
4
4
4
a
bbac
a
ac c
=
−−
=
ratio of the constant term and the
c. We can use b
a
− and ,
c
a which equal the
sum and product of the solutions of the
quadratic equation, respectively, to write
bc
2
a
=
()
2
21
0
32
xx
++=
⎜⎟
⎝⎠
2. a. 340
34
x
x
−=
=
Solution: 4
3
b. 2
2310
2, 3, 1
xx
abc
+−=
===−
() () ()( )
()
2
33421
22
x−± − −
=
()()
()
()
2
440
410
xx x
xx
−+− =
−+=
2
40 10
xx
xi
−= +=
=±
Solutions:
4, i±
() ()
()()
22
30 10
31
31
31
uu
uu
xx
xx
−= +=
==−
==−
=± =± −
()()
()( )
()
()()
32
22
120
1120
xx
xxxx
−−=
−++−=
205
() () ()()
114
2
13
x
=
−± −
=
−± −
=
±
f. 63
780xx−−=
() ()
()()
8 0 1 0
uu
−= +=
()
()
20 2xx−=⇒=
()
21
2416 212
x
=
−± − −± −
==
()
()
10 1xx+= ⇒=−
1, 1, 1
ab c
==−=
() () ()()
()
2
11411
21
x
−− ± − −
=
Solutions: 13
1, 2, 1 3, 2
i
i±
−−±
()
1
Let .
1
uy
=+
()( )
2
10 3 0
13
11 133
3
uu
uu
yy
+= −=
=− =
=− − = +
Let 1.un
=
() ()
()()
21
2310
nn
−−
++=
1
210 10
11
uu
u
un
−
+= +=
=−
=− =−
c. 13 16
340xx+−=
16
Let .ux=
() ()
()()
2
340
410
uu
uu
+−=
+−=
()
40 10
1
uu
x
+= −=
=
Solution: 1
Chapter 10 Quadratic Equations, Functions, and Inequalities
206
Equations
2. The graph of a quadratic function is called
4. For a parabola given by the function
()
()
a maximum point.
()
the axis of symmetry is .
2
b
xa
=−
() ( )
() ( )
() ( )
() ( )
() ( )
2
2
2
0200 0,0
12121,2
22282,8
−=
−=− −
−=− −
()
()
3
0000,0
3
=
()
()
() ( )
()
()
()
()
()
()
2
2
2
2
3,
3336 3,6
2231 2,1
11321,2
11321,2
xyfx x xy==−
−−−= −
−−−= −
−−−=−−−
−=− −
Section 10.4 Graphing Quadratic Equations 207
()
6
1, 6; 3
221
b
ab a
== −=−=−
() ()
6
-intercepts: 6,0 and 0,0
x
x
=−
−
() () ()
()
2
00600
-intercept: 0,0
f
y
=+ =
()
1, 4; 2
221
ab a
== −=−=−
()()
()
()
() () ()
()
2
004050
-intercept: 0, 5
f
y
=+−=
−
()
()
2
1, 2; 1
221
b
ab a
−
=− =− − =− =−
−
()()
40 20
42
xx
xx
+= −=
=− =
x-intercepts:
()
4, 0− and
()
2, 0
() () ()
()
2
002088
-intercept: 0, 8
f
y
=− − + =
22. 2
() 4fx x=−
2
40
x
−=
() ()
-intercepts: 2, 0 and 2,0
x
−
y-intercept:
()
0, 4
Chapter 10 Quadratic Equations, Functions, and Inequalities
208
24.
() ( )
121fx x x x=− =−+
()
1, 2
21
221
ab
b
a
==−
−
−=− =
()
() ( )
2
0011f=− =
y-intercept:
()
0,1
()
()
2
1, 6
63
221
ab
b
a
==
−=− =−
()
() () ()
()
2
006099
-intercept: 0,9
f
y
=++=
()
()
2
1, 1
11
2212
11 1 25
ab
b
a
==
−=− =−
⎛⎞⎛⎞⎛⎞
()()
30 20
xx
+= −=
x-intercepts:
()()
3, 0 2,0−
() () ()
()
2
00 066
-intercept: 0, 6
f
y
=+−=−
−
Section 10.4 Graphing Quadratic Equations 209
30.
() ()() ()
() ()() ()
2
2
2
2
22
Maximum
Function Opens Number of -intercepts Number of -intercepts
Minimum
4
06
00
0416 240
6Downward Maximum 1
No real solutions
0
400
00
204200
Upward Minimum 1
1
xy
bac
f
aa
fx x
bac f
aa
fx x
−
=−
<<
=−−−=−<
=− −
−=
>>
==−=
2
4
bac
−
()
()
()
() ( )
() ( )
() ( )
() ( )
2
2
2
2
2
212 2 7 2,7
112 1 1 1,1
01201 0,1
1121 1 1,1
2122 7 2,7
−−−=−−−
−−−=−−−
−=
−=− −
−=− −
()
() ()
()
2
00
222
0120 1
Vertex: 0,1
b
a
f
−=− =
−
=− =
Domain:
()
,;−∞ ∞ Range:
(
]
,1−∞
()
() ()
()
() () ( )
() () ( )
() () ( )
() () ( )
2
2
2
2
2
030 12066 0,6
131 1216 3 1,3
232 122 6 6 2,6
333 123 6 3 3,3
434 12466 4,6
−+=
−+=− −
−+=− −
−+=− −
−+=
()
() () ()
()
2
12 2
223
232 1226 6
Vertex: 2, 6
b
a
g
−
−=− =
=−+=−
−
Domain:
()
,;−∞ ∞ Range: [ 6, )−∞
Chapter 10 Quadratic Equations, Functions, and Inequalities
210
()
()
() ( )
() ( )
() ( )
() ( )
()
() ()
()
00
Vertex: 0, 2
b
−
()
(
]
()
() ()
()
() () ( )
() () ( )
() () ( )
2
2
2
2
555511 5,1
4454134,3
3353153,5
2252152,5
−−+−+= −
−−+−+=−−−
−−+−+=−−−
−−+−+=−−−
()
2
55
2212
Vertex: ,
24
b
a
−=− =−
⎛⎞
−−
⎜⎟
⎝⎠
4
⎣⎠
() () ( )
() () ( )
() ()
()
2
0303011 0,1
1313111 1,1
2323217 2,7
−+=
−+=
−+=
()
2
31
2232
11 1 1
b
a
−
−=− =
⎛⎞ ⎛⎞ ⎛⎞
Domain:
()
,;−∞ ∞ range: 1,
4
⎡⎞
∞⎟
⎢
⎣⎠
()
()
()()
Vertex: 0.55, 3.30
-intercepts: 1.27, 0 and 2.37, 0
x
−
−
()
()
()()
()
Vertex: 0.71,7.81
-intercepts: 1.65,0 and 3.08, 0
-intercept: 0 , 7.1
x
y
−
46.
()
()
2
328
Vertex: 0.33, 7.67
-intercepts: None
fx x x
x
=− + −
−
Section 10.4 Graphing Quadratic Equations 211
48.
() ( ) ( )() ()
() ()() ()
()
2
Downward Maximum 1
2
Upward Minimum 1
0
() ()() ()
() ()( ) ()
() ( )( ) ()
2
2
22
2
22
ard Minimum 1
1
402
00
2041280
Upward Minimum 1
2
405
00
653 6435 240
Downward Maximum 1
0
bac f
aa
fx x
bac f
aa
fx x x
−=−
>>
=− =− −=>
−=−
<<
=−− =−−−=−<
() ()
()
2
1, 4
42
22
224240
Vertex: 2,0
Axis of symmetry: 2
ab
b
a
f
x
==−
−
−=−=
=− +=
=
2
440
2
xx
x
−+=
=
()
-intercept: 2, 0x
()
()
-intercept: 0,4
y
()
() ()
()
() () ( )
() () ( )
() () ( )
() () ( )
2
2
2
2
2
12 1 4 1 39 1,9
020 4033 0,3
1214131 1,1
2224233 2,3
3234339 3,9
−−−−+=−
−+=
−+=
−+=
−+=
()
() () ()
()
2
121 4131
Vertex: 1,1
f
=−+=
Domain:
()
,;−∞ ∞ Range:
[
)
1, ∞
Chapter 10 Quadratic Equations, Functions, and Inequalities
212
()
()
11
b
−
() ( )
The minimum daily cost is $50.
300 2
lw
=−
b. Area = length × width
() ( )
()
2
300 2
300 2
A
www
A
www
=−
=−
c.
d.
()
() () ()
()
max
2
2 300
300 75
222
75 300 75 2 75 11, 250
300 2 300 2 75 150
ab
b
wa
R
lw
=− =
=− =− =
−
=−=
=−=− =
A pen measuring 75 ft by 150 ft will
produce a maximum area of 11,250 2
ft .
180
lw
=+
() ( )
180
A
ww w
=−
()
() ()
2
90 180 90 90 8100
A
=−=
The maximum possible area is 8100 ft2.
Mindstretchers
() ( )
() ( )
() ( )
() ( )
2
2
2
2
2231 2,1
11321,2
2231 2,1
−−−= −
−=− −
−=
()
23fx x=−
()
()
() ( )
() ( )
() ( )
() ( )
() ( )
2
2
2
2
2
2
1,
22132,3
1110 1,0
00110,1
11101,0
22132,3
xfx x xy=−
−−−= −
−−−= −
−=− −
−=
−=
()
21fx x=−
Section 10.4 Graphing Quadratic Equations
213
()
()
() ( )
() ( )
() ( )
() ( )
() ( )
2
2
2
2
2
2
1,
22152,5
11121,2
00110,1
11121,2
22152,5
xfx x xy=+
−−+= −
−−+= −
+=
+=
+=
()
21fx x=+
()
()
() ( )
() ( )
() ( )
() ( )
() ( )
2
2
2
2
2
2
2,
22262,6
11231,3
00220,2
11231,3
22262,6
xfx x xy=+
−−+= −
−−+= −
+=
+=
+=
()
22fx x=+
3:
c=−
The graph is shifted 3 units downward.
1:
c=−
The graph is shifted 1 unit downward.
1:
c=
The graph is shifted 1 unit upward.
2:
c=
The graph is shifted 2 units upward.
()
()
()
() ()
() ()
() ()
() ()
2
2
2
2
2
00390,9
1134 1,4
22312,1
3330 3,0
44314,1
−=
−=
−=
−=
−=
() ( )
2
3fx x=−
()
()
()
() ()
() ()
() ()
() ()
() ()
2
2
2
2
2
2
1,
1114 1,4
00110,1
1110 1,0
22112,1
33143,4
xfx x xy=−
−−−= −
−=
−=
−=
−=
() ( )
2
1fx x=−
()
()()
() ()
() ()
() ()
() ()
() ()
() ()
2
2
2
2
2
2
2
1,
33143,4
22112,1
11101,0
00110,1
11141,4
22192,9
xfx x xy=+
−−+= −
−−+= −
−−+= −
+=
+=
+=
() ( )
2
1fx x=+
Chapter 10 Quadratic Equations, Functions, and Inequalities
214
()
()
() ()
() ()
() ()
() ()
() ()
2
2
2
2
2
44244,4
33213,1
22202,0
11211,1
00240,4
−−+= −
−−+= −
−−+= −
−−+= −
+=
() ( )
2
2fx x=+
3:
c=−
The graph is shifted 2 units to the left.
downward.
iii. The graph is shifted c units to the
left.
2
2. 2
22 2
2
33 3
ax bx c y
++=
()()()
( ) () ()
( ) () ()
2
2
2
2,0 2 2 0
1, 4 1 1 4
1,6 1 1 6
abc
abc
abc
−−+−+=
−− − + − + =−
++=
42 0
4 4
abc
abc abc
−+=
−+=−⎯⎯→−+=−
()
410
ac
+=
()
1
54
1 1
ac
ac ac
×−
−+=−
+=⎯⎯→+=
3. The equation of the axis of symmetry is
() ()
The equation of the axis of symmetry is
()
33
,
221 2
b
xa
−
=− = =− which is equal to
352.yx x=−+
()
3
⎜⎟
⎝⎠
The equation of the axis of symmetry is
()
()
55,
2236
b
xa
−−
=− = = which is equal to
215
Section 10.5 Solving Quadratic and Rational Inequalities 215
Rational Inequalities
Exercises
2. A quadratic inequality in one variable is
()
2
40
40
xx
xx
+=
+=
0 4 0
4
xx
x
=+=
=−
() ()
2
45545550
x
<− − − + − = >
()()
8.
()()
2
9
x
>
30 30
3 3
xx
xx
+= −=
=− =
2
Test
Interval 9 Conclusion
x
>
()
()()
2150
530
xx
xx
+−=
+−=
50 30
53
xx
xx
+= −=
=− =
[]
12.
()
2
440
20, 2
xx
x
xx
−+≥
−= =
() ()
2
233434110
x
≥−+=≥
()
,−∞ ∞
14.
()()
20 0
540
xx
xx
−− ≤
+−=
50 40
5 4
xx
xx
+= −=
=− =
(
][
)
,5 4,−∞ − ∪ ∞
Chapter 10 Quadratic Equations, Functions, and Inequalities
216
16.
()( )
430
xx
−+≤
10 30
1 3
xx
xx
−= − =
==
2
Test
Interval 4 3 0 Conclusion
Value
xx
−+≤
[]
18.
()()
2
2520
2520
21 20
xx
xx
xx
++>
++=
++=
210 20
xx
+= + =
() ()
2121512110
2
x
−< <− − − + − + =− −≤
()
⎜⎟
20.
()()
2
940
940
32320
x
x
xx
−<
−=
+−=
320 320
3 2 3 2
xx
xx
+= −=
=− =
22.
2
2
4415
44150
xx
xx
+≥
+−=
22
()()
() ()
2
2
2
53
040400 015
22
324242242415
x
x
≤− − − + − = ≥
−≤≤ + = <
24. 10
x
1is undefined at 0
x
111
01 1 110
22
2
21 1 1
12 0
22 2
x
x
⎛⎞
⎟
⎜
<≤ − =− −≤
⎟
⎜⎟
⎜
⎝⎠
−
≥=>
217
22 2
30 0
30 3 3
x
x
−
<=≥
−
28.
()
66
64
6
64 6
64 24
xx
x
x
xx
xx
>=−
++
−=
+
−= +
−= +
06
60 114
x
−
>− =− − ≤
30. 22 2
32 32 3
20
32
20
2
xx
x
x
x
x
≥=−
++
−=
+
−=
=
32.
3 is undefined at
23 23 2
39
3
xx
x
≤=
−−
+=
()
()
()
()
30 9
30333
2203
33 9
363 6 63
2233
x
x
+
<=−−≤
−
+
<≤ = >
−
34.
1 is undefined at 5
55
31
x
xx
≥=
−−
Value 5
33 3
20 1
50 5 5
33 3
253 1
53 2 2
x
x
x
−
≤=<
−
≤< = ≥
−
218
2
340
xx
−+ +=
14
xx
=− =
() ()
2
2
2
Test Conclu-
Interval 3 4 0
Value sion
12 2324660
xx
x
−+ +≤
≤− − − − + − + =− − ≤
38. 11 3
()
()
Test 1
Interval 0 Conclusion
Value 23
21
32330
21 11
x
x
−≥
+
−−
−
()
()
16 0 3 0
03
tt
tt
−= −=
==
()
()
(
)
(
)
2
16 40 180 196
82 1 2 0
tt
tt
−++≥
−−−=
210 20
1 2
2
tt
tt
−= − =
==
() ()
16 3 40 3 180
−++
44. 2260
2602
60 2
wl
wl
l
w
+=
=−
−
=
()
2
10 20
ll
==
()
()
()
10 0 0 30 0 0 0 200
10 20 15 15 30 15 225 225 200
20 30 30 30 30 0 0 200
l
l
l
<−=<
<< − = ≥
>−=<
Section 10.5 Solving Quadratic and Rational Inequalities
219
46. 100 1800 10
100 1800 10
100 1800 10
90 1800
20
R
R
R
R
R
R
R
R
−=
−=
=
=
100 1800
Test
Interval 10 Conclusion
R
−>
1. The product of two numbers is negative when
the two numbers have the opposite signs.
Likewise, the quotient of two numbers is
negative when the two numbers have opposite
signs. So
()( )
130xx−+< when the values
of
(
)
(
)
1 and 3xx−+ have opposite signs,
2. To solve the compound inequality, solve the
satisfy both inequalities:
(
)
(
)
1, 1 2, 4 .−∪
2
2
2
436
436
320
xx
xx
xx
<−+
=−+
−+=
2
3610
3610
xx
xx
−+<
−+=
1 0 4 0
1 4
xx
xx
+= −=
=− =
() ()
() ()
2
2
2
Test
Interval 4 3 6 10 Conclusion
Value
1 3 3 3 3 6 24 24 10
110 030664610
xx
x
x
<−+<
<− − − − − + = ≥
−< < − + = < <
3. a. Strategies may vary. The boundary values
()()
280
240
xxx
xx x
−−≥
+−=
0 2 0 4 0
2 4
xx x
xx
=+= −=
=− =
32
Test
Interval 2 8 0 Conclusion
Value
xxx
x
−−≥
[][
)
3. b. Strategies may vary. The boundary values will
be where the quartic equation is zero. Check all
intervals created by the boundary values.
() ()
2
540
uu
−+=
220
3. b. (continued)
22
10 40
14
14
12
uu
uu
xx
xx
−= − =
==
==
=± =±
() ()
42
42
3 3 35 35
3
21 5 4 0
22 2 16 16
110 05044 40
x
x
⎛⎞ ⎛⎞
−< <− − − − − + =− − <
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
−≤ ≤ − + = ≥
()()
c. Strategies may vary. The boundary values
will be where the expression is zero
2
20 160
xx
−= − =
()
2
2
2
Test
Interval 0 Conclusion
Value 16
52 7 7
45 0
99
516
x
x
x
−≤
−
−−
<− − =− − ≤
−−
()
[
)
will be where the expression is zero
(numerator is zero) and the expression is
undefined (denominator is zero). Check all
intervals created by the boundary values.
() ()()
2
40
xx
−>
0 4 0
4
xx
x
=−=
=
20 30
2 3
xx
xx
+= +=
=− =
() ()
() ()
()()
2
2
2
141 55
201 0
44
116
3.5 3.5 6
x
−−−
−< ≤ − =− − ≤
−−−−
()()()