Chapter 11
Quadratic Equations and Functions
11.1 Check Points
2.
2
3110x
2
2
311
11
3
11
3
11 3
33
33
3
x
x
x
x
x
The solution set is
33 .
3
4.
2
(3)10x
310 or 3 10
310 310
xx
xx
The solution set is
310.
5. a.
2
10
xx
The coefficient of the x-term is 3.
Half of 3 is 3,
2
and
2
3
2
is 9
4 which
should be added to the binomial.
The result is a perfect square trinomial.
2
2
93
342
xx x
c.
2
3
4
xx
The coefficient of the x-term is 3.
4
Half of 3
2
3
25 or 2 5
25 25
xx
xx
The solution set is
25.
Chapter 11 Quadratic Equations and Functions
7.
2
2
2
2
2340
2340
2222
320
2
32
xx
xx
xx
xx
4
8.
2
2
2
3980
3980
3333
8
30
xx
xx
xx
2
2
2
989
3434
33227
21212
35
212
xx
x
x
interest rate. The solution is 0.2; the annual interest
rate is 20%.
10.
22 2
2
2
20 50
400 2500
2100
x
x
x
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
12.
1212
Midpoint ,
22
xxyy
11.1 Concept and Vocabulary Check
1. d
2. 7
3. 11
2
;
22
2
10. right; hypotenuse; legs
11. right, legs; the square of the length of the
hypotenuse
11.1 Exercise Set
1.
2
375
x
2.
2
2
520
4
x
x
44
or
22
xx
xx
The solution set is
2.
3.
2
2
742
x
2
25
16
x
Apply the square root property.
25
16
x
Chapter 11 Quadratic Equations and Functions
6.
2
2
449
49
4
x
x
7.
2
2
2
320
32
2
3
x
x
x
Apply the square root property.
8.
2
2
2
350
35
5
3
x
x
x
9.
2
2
2
25 16 0
25 16
16
x
x
4
5
44
055
xi
xii
The solution set is 4.
5i
4
7
2
7
02
xi
xi
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
2
13.
2
35x
Apply the square root property.
14.
2
43x
43 or 4 3
43 43
xx
xx
The solution set is
43.
15.
2
28x
Apply the square root property.
16.
2
2
3236
212
x
x
17.
2
59x
Apply the square root property.
2
The solution set is
52.i
2
311
311
44
311 311
44 4
x
x
The solution set is
311
.
4
20.
2
27
525
x
55
27
5
x
27
93
xx
The solutions are
9 and 3
and the solution set is
Chapter 11 Quadratic Equations and Functions
22.
2
2
6949
349
xx
x
23.
2
2 xx
24.
22
2
424
22
b
2
2
44 2xx x
26.
22
2
10 525
22
b
2
2
10 25 5xx x
27.
2
7 xx
Since b = 7, add
22
749
224
b
.
29.
2
1
2
xx
2222
31.
2
4
3
xx
Since
4
3
b
, add
32.
2222
44124
2
25 52525
b
2
2
44 2
525 5
xx x
33.
2
9
4
xx
9
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
35.
2
2
432
4 32
xx
xx
Since b = 4, add
22
2
424
b
.
36.
2
2
2
67
6979
316
xx
xx
x
37.
2
2
62
6 2
xx
xx
Since b = 6, add
22
2
639
22
b
.
38.
2
2
25
2 5
xx
xx
39.
2
2
810
8 1
xx
xx
Since
8b
, add
22
2
8416
b
.
40.
2
2
850
8 5
xx
xx
2
2
816516
xx
41.
2
220
2 2
xx
xx
Since b = 2, add
22
2
211
22
b
.
2
42.
2
2
480
48
4 8
xx
xx
xx
Chapter 11 Quadratic Equations and Functions
43.
2
2
310
3 1
xx
xx
Since b = 3, add
22
39
b
.
44.
2
2
2
350
35
3 5
xx
xx
xx
2
99
35
44
xx
45.
2
2
43
0
749
43
749
xx
xx
77
21
77
21 1 21 3
or
77 7 77 7
x
x
The solution set is
31
,
77
.
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
47.
2
2
10
1
xx
xx
Since b = 1, add
22
11
224
b
.
2
11
1
44
xx
48.
2
2
2
730
7 3
49 49
73
xx
xx
xx
49.
2
2
2
2350
35
0
22
35
22
xx
xx
xx
Since 3
b
, add
37 37
or
44 44
410
44
15
xx
xx
xx
57 3
44
x
or 57 1
44 2
x
The solution set is 1
3, .
2
Chapter 11 Quadratic Equations and Functions
51.
2
2
3610
1
20
xx
xx
Apply the square root property.
2
13
x
52.
2
3620xx
2
2
2
36 2
2
23
2
21 1
3
xx
xx
xx
53.
2
2
3810
81
0
xx
xx
2
39 39
4 3 16 13
3999
x
413413
33 3
x
The solution set is
413
3
.
54.
2
2340xx
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
55.
2
2
2
8410
11
0
28
11
28
xx
xx
xx
Since
1
2
b
, add
Apply the square root property.
44
The solution set is
11
44
i
.
56.
2
2
9650
65
0
99
xx
xx
2
25
39
xx
57.
2
2
2
2
2570
2570
2222
57
0
22
57
22
xx
xx
xx
xx
2
416
x
44
58.
2
2
2
2
4250
4250
4444
15
0
24
15
24
xx
xx
xx
xx
2
Chapter 11 Quadratic Equations and Functions
59.
2
9
() 25
29
525
gx
x
Apply the square root property.
The values are 1
5
and 1.
60.
2
1
3
gx x
3
61.
2
2
() 125
5( 2) 125
(2) 25
hx
x
x
Apply the square root property.
225
25
25
x
xi
xi
The values are
25
i
.
63.
22
22
14 2 8 3
12 5 144 25
169 13
d
66.
22
22
32 53 1 2 14
52.24
d
22
52 254
29 5.39
70.
22 2
2
24 31 6 2
36 4 40 4 10 2 10 6.32
d
71.
2
2
22
40 1 3
4 4 16 16
32 16 2 4 2 5.66
d
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
73.
22
22
3.5 0.5 8.2 6.2
42 164
20 4 5 2 5 4.47
d
76.
2
2
22
70 0 2
727293
d
77.
22
22
33 3 5 45
43 35
16 3 9 5 48 45
93 9.64
d
80.
22
22
22
31 61
44 77
47 11
d
22
12 10
,6,5
22
83.
2682
Midpoint ,
22
810
,4,5
22
84.
4173
Midpoint ,
22
510 5
Chapter 11 Quadratic Equations and Functions
87.
75311
222 2
Midpoint ,
88.
227 4
551515
Midpoint ,
22
43
41 3 1
515
,,
22 52152
43 21
,,
10 30 5 10
91.
18 2 4 4
Midpoint ,
22
92 2 0
,
22
92.
50 2 6 6
Midpoint ,
22
2
24
x
Apply the square root property.
242
22
xi
xi
The values are
22
i
and
22
i
.
94. Let x = the number.
2
Apply the square root property and keep only the
principal square root.
2
vgh
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
97.
2
2
1
1
A
Pr
Ar
P
A
98.
12
2
kP P
Cd
principal square root.
12
12 12
kP P
dC
kP P kP P C
C
dC
CC
99.
2
2
10
396
1
330
396
xx
xx
Apply the square root property.
119 19
636 6
x
2
13
0
32
xx
111 11
.
223 636
b
2
2
1131
336236
154155
6363636
xx
x
Apply the square root property.
155 55
636 6
155155
x
Chapter 11 Quadratic Equations and Functions
101.
22
22
2
2
xbx b
xbx b
Since
b
is the linear coefficient, add
22
.
24
bb
102.
22
22
6
6
xbxb
xbx b
Since b is the linear coefficient, add
22
.
24
bb
Apply the square root property.
2
25 5
242
5
22
bbb
x
bb
x
103. a.
2
8xx
b. 16
c.
2
816xx
Apply the square root property.
11.44
11.2
11.2
1 1.2 or 1 1.2
r
r
r
r
106.
2
2
1
2420 2000 1
1.21 1
t
AP r
r
r
11.21or1 1.21
1 1.1 1 1.1
0.1 2.1
rr
rr
rr
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
107.
2
2
1445 1280 1
1445 1
1280
r
r
108.
2
2
1
101, 250 80, 000 1
1.265625 1
t
AP r
r
r
109. a.
2
2
( ) 2 140
(20) 2(20) 140
940
fx x
f
According to the model there were 940
billionaires in 2007.
This underestimates the number displayed in the
graph by 6 billionaires.
110. a.
2
2
( ) 2 140
(25) 2(25) 140
1390
fx x
f
2
961
961
31
x
x
x
Reject 31.
According to the model, there will be 1940
300
10 3 17.3
t
t
Disregard –17.3 because we can’t have a negative
time measurement. The solution is 17.3 and we
conclude that the sky diver was in a free fall for
10 3 or approximately 17.3 seconds.
Chapter 11 Quadratic Equations and Functions
113.
x
3 miles
114.
22 2
42
x
115.
22 2
2
2
10 30
100 900
800
x
x
x
116. Ignoring the thickness of the tabletop, we
essentially need to find the diagonal of the
rectangular opening.
22 2
117.
50 feet
50 feet
222
118.
4 miles
2 miles
x
70 feet
x
Section 11.1 The Square Root Property and Completing the Square; Distance and Midpoint Formulas
9800
4900 2
x
x
119.
2
196 22 22
196 4 4
196 4
Alw
xx
xx
x
120.
2
225 4 4 4 4
225 8 8
225 16 64
Alw
xx
xx
xx
121. – 128. Answers will vary.
129.
2
410x
440
440
00,true
00,true
130.
2
190x
00,true 00,true
131. Answers will vary
132. does not make sense; Explanations will vary.
Sample explanation: After you take half of the b–
become
10 5
,or .
42
Thus you will add
2
525
,or ,
24
to both sides.
136. true
Chapter 11 Quadratic Equations and Functions
137. false; Changes to make the statement true will vary.
A sample change is:
2
512x
is equivalent to
523x
.
140.
22
22
1
xy
ab
22
22
1
yx
ba
141.
2
2
0
xxc
xx c
Since b = 1, add
22
11
.
224
b
Apply the square root property.
11
24
xc
141
24
114 114
22 2
c
cc
2
2
2
24
4
244
4
bb
xc
bcb
x
bcb
x