Chapter 10 Quadratic Equations, Functions, and Inequalities
204
Mindstretchers
1. a.
22
22
44
22
44
2
2
2
bb ac bb ac
aa
b b ac b b ac
a
bb
aa
−+ − −− −
+
−+ − − − −
=
−−
==
4
a
The product is .
c
a The product is the
ratio of the constant term and the
coefficient of the quadratic term.
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
2. a. 340
34
4
3
x
x
x
−=
=
=
Solution: 4
3
b. 2
2310
2, 3, 1
xx
abc
+−=
===−
d. 42
230xx−−=
2
Let .ux=
() ()
()()
2
22
2
230
230
310
xx
uu
uu
−−=
−−=
−+=
22
30 10
31
31
uu
uu
xx
−= +=
==−
==−
Section 10.3 More on Quadratic Equations
205
2. a. (continued)
() () ()()
2
11411
x
−± −
=
f. 63
780xx−−=
3
Let .ux=
() ()
()()
2
33
2
780
780
810
xx
uu
uu
−−=
−−=
−+=
33
8 0 1 0
80 10
uu
xx
−= +=
−= +=
3
80
x
−=
2
()
()
3
2
10
110
x
xxx
+=
+−+=
10 1xx+= ⇒=−
210
1, 1, 1
xx
ab c
−+=
==−=
2. ii. The number of solutions is less than or
equal to the degree of the polynomial.
11
⎛⎞
10 30
13
11
13
11
11 133
223
22
3
uu
uu
yy
yy
yy
yy
+= −=
=− =
=− =
++
=− − = +
=− − =
=− =−
Solutions: 2
2, 3
−−
b. 21
2310nn
−−
++=
2
n
=−
Solutions:
2, 1−−
c. 13 16
340xx+−=
16
Let .ux=
() ()
2
16 16
340
xx
+−=
Chapter 10 Quadratic Equations, Functions, and Inequalities
206
10.4 Graphing Quadratic
Equations
Exercises
2. The graph of a quadratic function is called
a parabola.
10.
()
2
2fx x=−
() ( )
() ( )
2
2
2,
222 8 2,8
xyfx x xy==−
−−−=− −−
12.
()
2
1
3
fx x=
()
()
()
()
2
2
2
1,
3
1
66126,12
3
1
xyfx x xy==
−−=−
14.
()
23fx x=−
Section 10.4 Graphing Quadratic Equations 207
16. 2
() 6fx x x=+
()
()() ()
()
2
6
1, 6; 3
221
33639189
Vertex: 3, 9
b
ab a
f
== −=−=−
−=− + −=− =−
−−
18.
()
245fx x x=+−
()
()() ()
()
2
4
1, 4; 2
221
224259
Vertex: 2, 9
Axis of symmetry: 2
b
ab a
f
x
== −=−=−
−=− + −−=−
−−
=−
20.
()
228fx x x=− − +
()
() () ()
()
2
2
1, 2; 1
221
112189
Vertex: 1,9
b
ab a
f
−
=− =− − =− =−
−
−=−− − −+=
−
22. 2
() 4fx x=−
()
() ()
2
0
1, 0; 0
221
040 4
b
ab a
f
=− = − =− =
−
=− =
Chapter 10 Quadratic Equations, Functions, and Inequalities
208
24.
() ( )
22
121fx x x x=− =−+
1, 2
21
ab
b
==−
−
−=− =
() ( )
2
0011f=− =
y-intercept:
()
0,1
26.
()
269fx x x=++
()
1, 6
63
221
ab
b
a
==
−=− =−
26. (continued)
()
2
1, 1
11
2212
11 1 25
6
22 2 4
125
Vertex: ,
24
ab
b
a
f
==
−=− =−
⎛⎞⎛⎞⎛⎞
−=− +−−=−
⎜⎟⎜⎟⎜⎟
⎝⎠⎝⎠⎝⎠
⎛⎞
−−
⎜⎟
⎝⎠
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
Maximum
Function Opens Number of -intercepts Number of -intercepts
Minimum
4
06
00
0416 240
6Downward Maximum 1
No real solutions
0
xy
bac
f
aa
fx x
−
=−
<<
=−−−=−<
=− −
1
32.
()
()
()
()
2
2
12 ,
212 2 7 2,7
xfx x xy=−
−−−=−−−
34.
()
()
() ()
()
2
2
3126 ,
030 12066 0,6
xgx x x xy=−+
−+=
Chapter 10 Quadratic Equations, Functions, and Inequalities
210
36.
()
()
()
()
2
2
20.5 ,
220.52 4 2,4
xfx x xy=− −
−−− −=− −−
Domain:
()
,;−∞ ∞ Range:
(
]
,2−∞ −
38.
()
()
() ()
()
2
2
51 ,
555511 5,1
xhxxx xy=++
−−+−+= −
()
2
55
2212
55 5 21
51
22 2 4
521
Vertex: ,
24
b
a
h
−=− =−
⎛⎞⎛⎞ ⎛⎞
−=− +−+=−
⎜⎟⎜⎟ ⎜⎟
⎝⎠⎝⎠ ⎝⎠
⎛⎞
−−
⎜⎟
⎝⎠
38. (continued)
Domain:
()
,;−∞ ∞ Range: 21,
4
⎡⎞
−∞
⎟
⎢
⎣⎠
()
2
31
2232
11 1 1
331
22 2 4
11
Vertex: ,
24
b
a
g
−
−=− =
⎛⎞ ⎛⎞ ⎛⎞
=−+=
⎜⎟ ⎜⎟ ⎜⎟
⎝⎠ ⎝⎠ ⎝⎠
⎛⎞
⎜⎟
⎝⎠
()
()()
()
Vertex: 0.55, 3.30
-intercepts: 1.27, 0 and 2.37,0
-intercept: 0, 3
x
y
−
−
−
44.
()
()
2
1.4 2 7.1
Vertex: 0.71, 7.81
fx x x
=− + +
Section 10.4 Graphing Quadratic Equations 211
48.
() ( ) ( )() ()
2
2
2
Maximum
Function Opens Number of –intercepts Number of –intercepts
Minimum
405
00
55 5 415450
Downward Maximum 1
2
xy
bac
f
aa
fx x x
−
=
<<
=− − + = − − − = >
50.
() ()
2
2
() 4 4
1, 4
42
22
224240
fx x x
ab
b
a
f
=−+
==−
−
−=−=
=− +=
()
-intercept: 0,4
y
52.
()
()
() ()
()
() () ( )
2
2
2
2
243 ,
12 1 4 1 3 9 1,9
0204033 0,3
xhx x x xy=−+
−−−−+=−
−+=
Chapter 10 Quadratic Equations, Functions, and Inequalities
212
54. a.
()
2
0.005 100Cx x x=−+
b.
56. a. 2300
300 2
lw
lw
+=
=−
b. Area = length × width
A
() ( )
300 2
A
www
=−
d.
()
max
2 300
300 75
222
ab
b
wa
=− =
=− =− =
−
58. a.
()
360 2
lw
=+
b. The maximum is located at the vertex.
1, 180ab=− =
()
180 90
221
b
a
−=− =
−
() ( )
() ( )
() ( )
2
2
2
2
11321,2
00330,3
11321,2
2231 2,1
−−−=−−−
−=− −
−=− −
−=
() ( )
() ( )
2
2
2
00110,1
11101,0
22132,3
−=− −
−=
−=
Section 10.4 Graphing Quadratic Equations
213
1. a. (continued)
()
()
() ( )
2
2
1,
22152,5
xfxx xy=+
−−+= −
()
()
() ( )
() ( )
2
2
2
2,
22262,6
11231,3
xfxx xy=+
−−+= −
−−+= −
3:c=−
The graph is shifted 3 units downward.
1:c=−
The graph is shifted 1 unit downward.
1:c=
1. b.
()
()()
()
()
2
2
2
3,
00390,9
xfx x xy=−
−=
()
()
()
() ()
() ()
2
2
2
1,
1114 1,4
00110,1
xfx x xy=−
−−−= −
−=
()
()()
() ()
() ()
() ()
2
2
2
2
1,
33143,4
22112,1
11101,0
xfx x xy=+
−−+= −
−−+= −
−−+= −
Chapter 10 Quadratic Equations, Functions, and Inequalities
214
1. b. (continued)
()
()
()
() ()
2
2
2,
44244,4
xfx x xy=+
−−+= −
3:c=−
The graph is shifted 3 units to the right.
1:c=−
The graph is shifted 1 unit to the right.
1:c=
The graph is shifted 1 unit to the left.
2:c=
The graph is shifted 2 units to the left.
c. i. The graph is shifted c units
upward.
d. The graph is shifted 2 units to the right
and 3 units downward.
2. 2
11 1
2
22 2
2
33 3
ax bx c y
ax bx c y
ax bx c y
++=
++=
++=
()
1
410 4 10
3 9
3
31
2
ac ac
a
a
c
c
×−
+= ⎯⎯⎯→− − = −
−=−
=
+=
=−
The equation is 2
352.yx x=+−
3. The equation of the axis of symmetry is
12
.
2
xx
x+
=
Consider the equation 2
352.yx x=−+
The x-intercepts are
()
2,0 and 1,0 .
3
⎛⎞
⎜⎟
⎝⎠
The equation of the axis of symmetry is
Section 10.5 Solving Quadratic and Rational Inequalities 215
10.5 Solving Quadratic and
Rational Inequalities
Exercises
2. A quadratic inequality in one variable is
an inequality that contains a quadratic
expression.
() ()
() ()
() ()
2
2
2
2
Test
Interval 4 0 Conclusion
Value
45545550
401141330
0224212120
xx
x
x
x
+>
<− − − + − = >
−< < − − + − =− −<
>+=>
()
()
()
2
2
2
2
Test
Interval 9 Conclusion
Value
35525925
330 00 90
34416916
x
x
x
x
>
<− − − = <
−< < = >
>=<
10.
()()
2
2
2
278
2150
2150
530
xx
xx
xx
xx
+−≤
+−≤
+−=
+−=
50 30
53
xx
xx
+= −=
=− =
2
Test Conclu-
Interval 2 7 8
Value sion
xx
+−≤
()
2
2
440
20
20, 2
xx
x
xx
−+=
−=
−= =
2
2
Test
Interval 4 4 0 Conclusion
Value
xx
−+≥
5 4
xx
=− =
()()
() ()
2
2
2
Test Conclu-
Interval 20 0
Value sion
5 6 20 6 6 10 10 0
540 200020200
xx
x
x
−− ≤
≤− − − − − − =− − ≤
−≤ ≤ − − = >
Chapter 10 Quadratic Equations, Functions, and Inequalities
216
16.
()( )
2
2
430
430
130
xx
xx
xx
−+≤
−+=
−−=
18.
()()
2
2
2520
2520
21 20
xx
xx
xx
++>
++=
++=
210 20
2 1 2
1
2
xx
xx
x
+= + =
=− =−
=−
2
Test Conclu-
Interval 2 5 2 0
Value sion
xx
++>
20.
()()
2
2
940
940
32320
x
x
xx
−<
−=
+−=
320 320
xx
+= −=
20. (continued)
22
,
33
⎛⎞
−
⎜⎟
⎝⎠
()()
() ()
() ()
2
2
2
Value
5343 4324 2415
2
53
040400015
22
324242242415
2
x
x
x
≤− − − + − = ≥
−≤≤ + = <
≥+=≥
53
,,
22
⎛⎤⎡⎞
⎟
⎜⎥⎢
−∞ − ∪ ∞⎟
⎜⎟
⎜⎥⎢
⎝⎠
⎦⎣
01 2 20
1
111
01 1 110
22
2
21 1 1
12 0
x
x
x
<− = >
−
⎛⎞
⎟
⎜
<≤ − =− −≤
⎟
⎜⎟
⎜
⎝⎠
−
≥=>
Section 10.5 Solving Quadratic and Rational Inequalities
217
26. 20
3
20
3
20
x
x
<
−
=
−
≠
2is undefined at 3
3x
x=
−
Test 2
Interval 0 Conclusion
Value 3
x
<
−
28.
66
4 is undefined at 6
66
xx x
xx
−−
>=−
++
Test 6
Interval 4 Conclusion
Value 6
20 6 13 13
10 20 4
20 6 7 7
x
x
x
−>
+
−−
<− − = ≤
−+
30. 22 2
0 is undefined at
32 32 3
20
32
xx x
xx
x
x
−−
≥=−
++
−=
+
30. (continued)
[)
2
,2,
3
⎛⎞
⎟
⎜−∞ − ∪ ∞
⎟
⎜⎟
⎜
⎝⎠
32.
39 39 3
3 is undefined at
xx x
++
≤=
()
Test 39
Interval 3 Conclusion
Value 23
30 9
30333
x
x
+≤
−
+
[)
3
,6,
2
⎛⎞
⎟
⎜−∞ ∪ ∞
⎟
⎜⎟
⎜
⎝⎠
2
x
=
Test 3
Interval 1 Conclusion
Value 5
33 3
20 1
50 5 5
x
x
≥
−
≤=<
−
Chapter 10 Quadratic Equations, Functions, and Inequalities
218
36.
2
2
340
340
xx
xx
−+ +≤
−+ +=
() ()
2
2
Test Conclu–
Interval 3 4 0
Value sion
12 2324660
xx
x
−+ +≤
≤− − − − + − + =− − ≤
38. 11 3
0 is undefined at
23 23 2
10
23
10
1
xx x
xx
x
x
x
x
−−
≥=−
++
−=
+
−=
=
Test 1
Interval 0 Conclusion
Value 23
x
x
−≥
+
40.
()
2
16 48 0ht t t=− + >
()
2
16 48 0
16 3 0
tt
tt
−+=
−−=
42.
()
2
16 40 180ht t t=− + +
(
)
(
)
2
82 1 2 0
tt
−−−=
210 20
1 2
2
tt
tt
−= − =
==
() ()
2
21 204196
2204
16 3 40 3 180
2 3 156 196
156
t
t
≤≤ ≥
=
−++
><
=
The ball is at least 196 ft above the ground
between, and including, ½ sec and 2 sec.
44. 2260
2602
wl
wl
+=
=−
10 0 20 0
10 20
ll
ll
−= −=
==
()
Test
Interval 30 200 Conclusion
Value
ll
−≥
Section 10.5 Solving Quadratic and Rational Inequalities
219
46. 100 1800 10
100 1800 10
R
R
R
R
−>
−=
Mindstretchers
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
solutions.
2. To solve the compound inequality, solve the
individual inequalities 2
436 andxx<−+
23610.xx−+< The solutions of the
compound inequality are all values of x that
2. (continued)
1 0 4 0
1 4
xx
xx
+= −=
=− =
3. a. Strategies may vary. The boundary values
will be where the cubic equation is zero.
Check all intervals created by the boundary
values.
()()
32
280
240
xxx
xx x
−−≥
+−=
0 2 0 4 0
xx x
=+= −=
() () ()
() () ()
32
32
12181
041 90
9
52585
45 350
35
x
x
−−
<< −<
=−
−−
≥≥
=
Chapter 10 Quadratic Equations, Functions, and Inequalities
220
3. b. (continued)
22
10 40
14
14
12
uu
uu
xx
xx
−= − =
==
==
=± =±
() ()
42
42
Test Conclu–
Interval 5 4 0
Value sion
2 3 3 5 3 4 40 40 0
xx
x
−+<
≤− − − − − + = ≥
c. Strategies may vary. The boundary values
will be where the expression is zero
(numerator is zero) and the expression is
undefined (denominator is zero). Check all
intervals created by boundary values.
2
20
16
x
x
−≤
−
2
20 160
xx
−= − =
3. d. Strategies may vary. The boundary values
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
2
22
40
6
40 60
xx
xx
xx xx
−>
−−
−= −−=
() ()
() ()
() ()
() ()
() ()
()()
2
2
2
2
2
2
2
141 55
201 0
44
116
141 11
031 0
22
116
3.5 4 3.5 77
343.5 0
11 11
3.5 3.5 6
x
x
x
−−−
−< ≤ − =− − ≤
−−−−
−
<< = >
−−
−
<≤ =− − ≤
−−