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7/31/2019 7146992 4ACh01Quadratic Equations in One Unknown
1/32
Certificate Mathematics in Action Full Solutions 4A
1 Quadratic Equations in One Unknown
Activity
Activity 1.1 (p. 28)
1.
2.
3. 3, 2
4.
0)2)(3(
062
=+=+
xx
xx
2or3
02or03
====+
xx
xx
5. The quadratic equation ax2 + bx + c = 0 can be solvedgraphically by reading thex-intercepts of the graph of
y = ax2 + bx + c.
Follow-up Exercise
p. 3
1. Let ,8.0 =x
(2)888888.810
(1)888888.0i.e.
==
x
x
9
880.
9
8
89,(1)(2)
=
=
=
x
x
2. Let ,61.0 =x
(2)666666.110
(1)666166.0i.e.
==
x
x
6
160.1
6
1
90
15
1590
5.19,(1)(2)
=
=
=
==
x
x
x
3. Let ,21.0 =x
(2)212121.12100
(1)212121.0i.e.
==
x
x
33
4210.
33
4
99
121299,(1)(2)
=
=
==
x
x
4. Let ,321.0 =x
(2)123123.1231000
(1)123123.0i.e.
==
x
x
333
413210.
333
41
999
123
123999,(1)(2)
=
=
=
=
x
x
p.6
Quadratic equation General formax2 + bx + c = 0
Value of
a b c
(a) 5x2 = 6 x 5x2 +x 6 = 0 5 1 6(b) x2 4 = 5x x2 5x 4 = 0 1 5 4
(c) 3x2 = 4 3x2 + 0x 4 = 0 3 0 4
(d) 8x=x2 x2 8x + 0 = 0 1 8 0
(e) 2x(3 x) = 0 x2 3x + 2 = 0 1 3 2
(f) (1+x)(1 x) + 3x = 2 x2 3x + 1 = 0 1 3 1
p.8
1. (a) x2 10x + 16 = 0(x 2)(x 8) = 0
8or2
08or02
====
xx
xx
(b) 2x2 + 13x + 15 = 0
1
x 4 3 2 1 0 1 2 3
x2 16 9 4 1 0 1 4 9
x 4 3 2 1 0 1 2 3
6 6 6 6 6 6 6 6 6
y 6 0 4 6 6 4 0 6
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1 Quadratic Equations in One Unknown
(2x + 3)(x + 5) = 0
5or2
3
05or032
==
=+=+
xx
xx
(c)
0)2)(13(
0253 2
=+
=
xx
xx
2or3
1
02or013
==
==+
xx
xx
2. (a)
0)23)(34(
0612 2
=+=+
xx
xx
3
2or
4
3
023or034
==
==+
xx
xx
(b)
0)52)(52(
0254254
2
2
=+==
xx
xx
2
5or
2
5
052or052
==
==+
xx
xx
(c)
0)3)(5(
0152
152
15)2(
2
2
=+=+
=+
=+
xx
xx
xx
xx
3or503or05
== ==+ xxxx
(d)
0)3)(1(
034
6232
)3(2)3)(1(
2
2
==+
=
=+
xx
xx
xxx
xxx
3or1
03or01
====
xx
xx
Alternative Solution
0)1)(3(
0]2)1)[(3(
0)3(2)3)(1(
)3(2)3)(1(
==+ =+
=+
xx
xx
xxx
xxx
1or3
01or03
====
xx
xx
p.10
1. (a)
.0134isequationrequiredThe
0134
0)14)(1(
014or01
04
1or01
4
1or1
2
2
=+
=+
=+==+
==+
==
xx
xx
xx
xx
xx
xx
(b) 2
1or
3
2 == xx
012or023
02
1or0
3
2
=+=
=+=
xx
xx
026
0)12)(2(3
2 =
=+
xx
xx
The required equation is 6x2 x 2 = 0.
2. (a)
0)8)(12(08152 2
=+=
xx
xx
8or2
1
08or012
==
==+
xx
xx
(b) The roots of the required equation are
2
1
1
and 81
i.e. 2 and8
1.
018or02
08
1or02
8
1or2
==+
==+
==
xx
xx
xx
02158
0)18)(2(
2 =+
=+
xx
xx
The required equation is 8x2 + 15x 2 = 0.
3. (a)
0)4)(34(
012134 2
=+=
xx
xx
4or
4
3
04or034
==
==+
xx
xx
(b) The roots of the required equation are
24
3and
2
4, i.e.
8
3 and 2.
02or0
8
3
2or8
3
==+
==
xx
xx
8x + 3 = 0 or x 2 = 0
06138
0)2)(38(
2 =
=+
xx
xx
The required equation is 8x2 13x 6 0.
2
7/31/2019 7146992 4ACh01Quadratic Equations in One Unknown
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Certificate Mathematics in Action Full Solutions 4A
p.13
1. (a)
31
9)1( 2
=+=+
x
x
231
=+=x
oror
431
(b)
232
4)32( 2
==
x
x
or2
5
or232
=
+=
x
x
2
1
23
2. (a)
52
5)2( 2
=
=
x
x
52or52 +=x
(b)
64
1
64
12
=+
=
+
x
x
64
1or6
4
1 +=x
3. (a)
83
8)3( 2
=
=
x
x
d.p.)2to(cor.0.17ord.p.)2.to(cor.83.5
83or83
=+=x
(b)4
5)2( 2 =+x
)d.p.2tocor.(12.3or)d.p.2tocor.(88.0
2
52or
2
52
2
52
4
52
=
+=
=+
=+
x
x
x
p. 15
1. 22
2 )9(2
1818 +=
++ xxx
2. 22
2 )6(2
1212 =
+ xxx
3.
22
2
2
7
2
77
+=
++ xxx
4. 22
2 )4(2
88 +=
++ xxx
5.22
2
29
299 =+
xxx
6.22
2
4
1
4
1
2
1
=
+ xxx
7.22
2
6
1
6
1
3
1
=
+ xxx
8.22
2
4
5
4
5
2
5
+=
++ xxx
p. 16
1. (a)
1or9
54
5425)4(
2
89
2
88
98
098
2
22
2
2
2
==
==
+=
+
=
=
x
x
xx
xx
xx
xx
(b)
2
1or3
4
7
4
5
4
7
4
5
16
49
4
5
4
5
2
3
4
5
2
5
2
3
2
5
02
3
2
5
0352
2
22
2
2
2
2
=
=
=
=
+=
+
=
=
=
x
x
x
xx
xx
xx
xx
2. (a)
2
537or
2
537
2
53
2
7
2
53
2
7
4
53
2
7
2
71
2
77
17
017
2
222
2
2
+=
=
=+
=
+
+=
++
=+
=+
x
x
x
xx
xx
xx
3
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1 Quadratic Equations in One Unknown
(b)
3
51or
3
51
3
51
3
5)1(
2
2
3
2
2
2
2
3
22
03
22
0263
2
222
2
2
2
+=
=
=
+=+
=
=
=
x
x
x
xx
xx
xx
xx
p.20
1. Using the quadratic formula,
8or4
2
124
2
1444
)1(2
)32)(1(444
24
2
2
=
=
=
=
=a
acbbx
2. Using the quadratic formula,
2
1or2
4
35
4
95
)2(2
)2)(2(4)5()5(
24
2
2
=
=
=
=
=a
acbbx
3.
08103
8103
2
2
=+
+=
xx
xx
Using the quadratic formula,
4or32
6
1410
6
19610
)3(2
)8)(3(41010
2
4
2
2
=
=
=
=
=a
acbbx
4. Using the quadratic formula,
134or134
2
1328
2
528
)1(2
)3)(1(488
2
4
2
2
+=
=
=
=
=a
acbbx
5. 34 2 += xx0342 =+ xx
Using the quadratic formula,
72or72
2
724
2
284
)1(2
)3)(1(444
2
4
2
2
+=
=
=
=
=a
acbbx
p. 25
1.
2. The equationx2 9x + k= 0 has a double real root.
4
81
0481
0))(1(4)9(
0
2
=
==
=
k
k
k
3. The equation (2k 1)x2 + 3x 6 = 0 has no realroots.
4
Quadratic equations Value ofx2 + 4x + 2 = 0 = 42 4(1)(2) = 8 > 02x2 3x 5 = 0 = (3)2 4(2)(5) = 49 > 0x2 + 8x + 16 = 0 = 82 4(1)(16) = 02x2 + 5x = 0 = 52 4(2)(0) = 25 > 04x2 + 25 = 0 = 02 4(4)(25) = 400 < 0
Nature of roots
2 distinctreal roots
1 double real root No real roots
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Certificate Mathematics in Action Full Solutions 4A
16
548
15
01548
024489
0)6)(12(43
0
2
>
k
k
k
The range of possible values ofkis k< 89
.
10. The equation 5x2 + 3x + (k+ 1) = 0 has two distinct
real roots.
20
11
02011020209
0)1)(5(43
0
2
>
>+
>
k
kk
k
The range of possible values ofkis k< .2011
11. The quadratic equation 3x2 + 4x + k = 0 has real
roots.
0
i.e. 0))(3(442 k
3
4
1612
01216
k
k
k
3
4isofvaluestheofrangeThe kk
.
12. The quadratic equation (k+ 1)x2 2kx + (k 2) = 0
has real roots.
0
i.e.0)2)(1(4)2( 2 + kkk
2
84
084
08444 22
+++
k
k
k
kkk
The range of the values ofkis k2.
15
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1 Quadratic Equations in One Unknown
13. The equation 3x2 + 5xk= 0 has no real roots.
12
25
01225
0))(3(45
0
2
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Certificate Mathematics in Action Full Solutions 4A
(b)
16. (a) two
(b) Thex-intercepts ofy = 4x2 4x 3 are 0.5 and1.5.
Therefore, the roots of the equation 4x2 4x 3 = 0are 0.5 and 1.5.
17. (a) one
(b) Thex-intercepts ofy =x2 6x + 9 is 3.Therefore, the roots of the equationx2 6x = 9 is3.
18. Letx be one of the number, then 27 x is the othernumber.
0)12)(15(
018027
18027
180)27(
2
2
==+
=
=
xx
xx
xx
xx
15015
== xx
or
or
12012
== xx
The two numbers are 12 and 15.
Alternative Solution
Letx be one of the number, thenx
180is the other
number.
0)12)(15(
018027
27180
27180
2
2
==+
=+
=+
xx
xx
xx
xx
15
015
==
x
xor
or
12
012
==
x
x
The two numbers are 12 and 15.
19. Letx cm be the length of the rectangle, then
cm)19(2
238x
x = is the width of the rectangle.
0)8)(11(
08819
8819
88)19(
2
2
==+
=
=
xx
xx
xx
xx
11
011
==
x
x
or
or
8
08
==
x
x
The length and width of the rectangle are 11 cm and8 cm respectively.
Alternative Solution
Letx cm be the length of the rectangle, thenx
88cm is
the width of the rectangle.
0)8)(11(
08819
1988
1988
3888
2
2
2
==+
=+
=+
=
+
xx
xx
xx
xx
xx
11
011
==
x
x
or
or
8
08
==
x
x
The length and width of the rectangle are 11 cm and 8 cmrespectively.
20.
0)32)(12(
0384
348
2
2
==+
=
xx
xx
xx
5.0
012
==
x
xor
or
5.1
032
==
x
x
After 0.5 seconds and 1.5 seconds, the ball is 3 mabove the ground.
21. The equationx2ax 40 = 0 has two distinct realroots.
160
0160
0)40)(1(4)(
0
2
2
2
>
>+
>
>
a
a
a
The square of any numbers is always positive.
The equationx2ax 40 = 0 has two distinct realroots for any real values ofa.
a = 1 or 2 or 3 (or any other reasonable answers)
22.
09
1
3
44
3
12
2
2
=
+
=
mxx
mx
Using the quadratic formula,
8
m163
4
)4(2
9
1)4(4
3
4
3
4
2
4
2
2
=
=
=
m
a
acbbx
To have two rational roots of different signs, we need
9
1
9
16m16
3
4m16
>
>
>
m
25
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1 Quadratic Equations in One Unknown
m = 1 or 9 or 16 (or any other reasonable answers)
23. The graph ofy = ax2 + 4x + c intersects thex-axis atone point.
The equation ax2 + 4x + c = 0 has a double realroot.
4
0416
04)4(0
2
=== =
ac
ac
ac
a = 2, c = 2 ora = 1, c = 4 ora = 2, c = 2.(or any other reasonable answers)
24. The graphy = x2 + 2x + kdoes not intersect thex-axis.
The equation x2 + 2x + k= 0 has no real roots.
1
044
0))(1(42
0
2
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Certificate Mathematics in Action Full Solutions 4A
9
821or
9
821
18
822
18
3282
)9(2
)9)(9(4)2()2(
2
4
2
2
+=
=
=
=
=
a
acbbx
31.
0)3)(12(
0352
44816
)12)(1(4)13)(12(
2
22
=+=
=+
+=+
xx
xx
xxxx
xxxx
3or2
1
03or012
==
==+
xx
xx
Alternative Solution
0)3)(12(
0)3)(12(
0)]1(4)13)[(12(
0)12)(1(4)13)(12(
)12)(1(4)13)(12(
=+=++=+=++
+=+
xx
xx
xxx
xxxx
xxxx
3or2
1
03or012
==
==+
xx
xx
32.
1
01
0)1(
012
044412
04)1(4)1(
2
2
2
2
==+=+
=++
=+++
=++
x
x
x
xx
xxx
xx
Alternative Solution
[ ]
1
01
0)1(
02)1(
04)1(4)1(
2
2
2
==+=+=+
=++
x
x
x
x
xx
33. (a)
0)4)(32(
012112 2
==+
xx
xx
4or2
3
04or032
==
==
xx
xx
(b) The roots of the required equation are
2
32 and
2(4), i.e. 3 and 8.
02411
0)8)(3(
08or03
8or3
2 =+
=====
xx
xx
xx
xx
The required equation is 024112 =+ xx .
34. (a)
0)3)(13(
0383 2
=+=+
xx
xx
3or3
1
03or013
==
=+=
xx
xx
(b) The roots of the required equation are
3
11
and3
1
.
i.e. 3 and3
1 .
0383
0)13)(3(
013or03
03
1or03
3
1or3
2
=
==+=
=+=
==
xx
xx
xx
xx
xx
The required equation is 0383 2 = xx .
35. (a) The equation 016)5(22 =+ xkx has adouble real root.
1or945
45
16)5(
016)5(
064)5(4
0)16)(1(4)]5(2[
0
2
2
2
2
= =
==
=
=
=
=
k
k
k
k
k
k
(b) For 9=k ,
4
04
0)4(
0168
016)59(2
2
2
2
===
=+
=+
x
x
x
xx
xx
For 1=k ,
27
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1 Quadratic Equations in One Unknown
4
04
0)4(
0168
016)51(2
2
2
2
==+=+
=++
=+
x
x
x
xx
xx
36. (a) The equation 0)2(32 =+ kxx has realroots.
4
17
0174
0)2(49
0)]2()[1(4)3(
0
2
++++
k
k
k
k
The range of possible values ofkis4
17k .
(b)4
17k
Minimum value ofkis4
17 .
(c)
2
3
023
02
3
04
93
024
173
2
2
2
=
=
=
=+
=
+
x
x
x
xx
xx
37. (a)
0)2(44
244
2
2
=+
=+
kxx
kxx
The equation 0)2(44 2 =+ kxx has noreal roots.
3
048160)2(1616
0)2)(4(4)4(
0
2
>
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Certificate Mathematics in Action Full Solutions 4A
(b) (i) 1 double real root
(ii) Thex-intercept of 1682 += xxy is 4.Therefore, the root of the equation
16)8( =xx is 4.
41. The graph of kxkxy ++= 52 touches thex-axis..
The equation 052 =++ kxkx has a double realroot.
0)25)(25(
0425
0))((45
0
2
2
=+=
=
=
kk
k
kk
2
5or
2
5
025or025
==
==+
kk
kk
42. The graph of 1322
+= kxxy has two distinctx-intercepts.
The equation 0132 2 =+ kxx has two distinctreal roots.
8
1
081
0189
01243
0
2
>+>
>
k
k
)(k
)k)(()(
The range of possible values ofkis8
1
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1 Quadratic Equations in One Unknown
Area ofABC
2
2
2
cm180
cm2
409
cm2
)267()36(
2
=
=
+=
= BCAC
Perimeter ofABC
cm90
cm)94041(
cm)]36()267()566[(
=++=
++++=++= ACBCAB
46. (a) Area of shaded region
22
22
2
cm)5015(
cm)48215(
cm82
34)13)(25(
=
=
+=
xx
xx
xx
(b) (i) 1865015 2 =xx 023615 2 = xx
(ii)
0)4)(5915(
023615 2
=+=
xx
xx
4or(rejected)15
59
04or05915
==
==+
xx
xx
cm5
cm]44)143[(
=+=
= KCBJBCJK
cm12
cm]33)245[(
==
= HBAGABGH
Multiple Choice Questions (p. 48)
1. Answer: C
0)2)(32(062
2
=+=
xxxx
2or2
3
02or032
==
==+
xx
xx
2.
Answer: B
2
2
)14(
1816
+=
++
x
xx
3. Answer: D
Thex-intercepts of the graph are 5 and 2. The roots of the graph are 5 and 2.
02or05
2or5
==+==
xx
xx
0103
0)2)(5(
2 =+
=+
xx
xx
The required equation is 1032 += xxy .
4. Answer: D
For 0322 =++ xx ,
8
)3)(1(42 2
==
0 < The equation 0322 =++ xx has no real roots.
The graph 322 ++= xxy has no x-intercepts.For 0=x
3
3)0(202
=++=y
The graph 322 ++= xxy has positivey-intercept.
5. Answer: B
The graph cxxy += 42 touches thex-axis.
The equation 042 =+ cxx has a double realroot.
4
0416
0))(1(4)4(
0
2
===
=
c
c
c
6. Answer: AThe equation 08)8(
2 =+++ kxkx has a doubleroot.
8
08
0)8(
06416
0326416
0)8)(1(4)8(
0
2
2
2
2
===
=+
=++
=+
=
k
k
k
kk
kkk
kk
7. Answer: A
The equation 082 =+ pxx has no real roots.
16
0464
0))(1(4)8(
0
2
>
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Certificate Mathematics in Action Full Solutions 4A
5or2
05or02
====
xx
xx
9. Answer: B
is a root of 0532 2 =+ xx .
5
5)0(2
5)532(2564
0532
22
=+=
++=+
=+
.
10. Answer: A
The graph cxxy += 82 has twox-intercepts.
The equation 082 =+ cxx has two distinct realroots.
16
0464
0))(1(4)8(
0
2
>
>
c
c
c
The range of possible values ofc is c < 16.
HKMO (p. 49)
04)4()241(
0)2(4441
0)()(
2
22
=+
=+++
=+
xkxk
xxkxx
xkgxf
The equation 0)()( =+ xkgxf has a single root.
0)40)(16(
064024
032656168
0)4)(241(4)]4([
0
2
2
2
=+=+
=++
==
kk
kk
kkk
kk
40or16
040or016
===+=
kk
kk
40=d
Lets Discuss
p. 20
Angels method:
04
1
3
2
2
5
4
1
3
2
2
5
2
2
=
=
xx
xx
Using the quadratic formula,
30
1064or
30
1064
30
106
15
2
5
18
53
3
2
2
52
4
1
2
54
3
2
3
22
+=
=
=
=x
Kens method:
03830
3830
4
1
3
2
2
5
2
2
2
=
=
=
xx
xx
xx
Using the quadratic formula,
30
1064or
30
1064
30
1064
60
10628
60
4248
)30(2
)3)(30(4)8()8( 2
+=
=
=
=
=x
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