Chapter 7 Exercise 7.1
Q. 1.
(x + 3)(x + 4) = 0
x = −3
Q. 2.
(x + 1)(x + 9) = 0
or x = −2
or x = −9
(x − 9)(x + 8) = 0
Q. 9.
(5x + 11)(x − 3) = 0
11
x = − __ or x = 3
5
Q
Q. 18.
4x2 − 15x + 9 = 0
(4x − 3)(x − 3) = 0
3
x = __ or x = 3
4
49y2 − 225 = 0
Q.
Q 19.
(7y)2 − (15)2 = 0
or x = −8
(7y − 15)(7y + 15) = 0
2x2 − 5x + 2 = 0
15
y = ± __
7
Q. 20.
Q
x(2x + 3) = 0
3
or x = − __
2
x=0
or x = −1
Q. 17.
Q
or x = 3
(2x − 1)(x − 2) = 0
1
x = __ or x = 2
2
Q. 8.
x=5
(x + 5)(x − 3) = 0
x=9
Q. 7.
(x − 5)(x + 1) = 0
(x + 8)(x + 9) = 0
x = −5
Q. 6.
x2 − 4x − 5 = 0
or x = −9
x = −8
Q. 5.
Q.
Q 16.
(x − 5)(x + 2) = 0
x=5
Q. 4.
(2x − 5)(x + 9) = 0
5
x = __ or x = −9
2
or x = −4
x = −1
Q. 3.
Q 15.
Q.
Q. 21.
Q
(7x + 2)(x − 6) = 0
2
x = − __ or x = 6
7
x2 − 11x = 0
x(x − 11) = 0
x2 = 81
x=0
x = ±9
Q
Q. 22.
or x = 11
45x2 − 4x − 1 = 0
Q. 10.
(5x + 4)(x + 3) = 0
4
x = − __ or x = −3
5
Q. 11.
(5x + 3)(x + 6) = 0
3
x = − __ or x = −6
5
Q. 23.
Q
Q. 12.
(7x − 3)(x − 3) = 0
3
x = __ or x = 3
7
(5x − 1)(x + 7) = 0
1
x = __ or x = −7
5
Q.
Q 24.
x2 − 2x + 1 = 0
Q. 13.
(x − 1)2 = 0
x2 − 3x = 0
x(x − 3) = 0
x=0
Q. 14.
(9x + 1)(5x − 1) = 0
1
1
x = − __ or x = __
5
9
x2
or x = 3
x=1
Q.
Q 25.
150x2 + 5x − 2 = 0
−1=0
x2
(15x + 2)(10x − 1) = 0
2
1
x = − __ or x = __
15
10
=1
x = ±1
75x2 + 2.5x − 1 = 0
Q. 26.
Q
(5x + 6)(x − 4) = 0
6
x = − __ or x = 4
5
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
1
Q. 27.
Q. 28.
Q. 29.
(2x + 3)(x − 4) = 0
3
x = − __ or x = 4
2
(3x + 8)(x − 1) = 0
8
x = − __ or x = 1
3
x = ±2
Q. 41.
Q
Q.
Q 42.
Q. 31.
Q. 32.
(7x + 2)(x − 6) = 0
2
x = − __ or x = 6
7
Q. 43.
Q
(7x − 3)(2x + 3) = 0
3
3
x = __ or x = − __
7
2
Q. 44.
Q
3x2 + 14x + 8 = 0
(3x + 2)(x + 4) = 0
2
x = − __ or x = −4
3
x2 − 17x + 42 = 0
(x − 3)(x − 14) = 0
x=3
Q. 33.
Q. 34.
or x = 14
Q.
Q 45.
8x2 − 54x + 13 = 0
(4x − 1)(2x − 13) = 0
1
13
x = __ or x = ___
2
4
(6x + 1)(x + 2) = 0
1
x = − __ or x = −2
6
3x2 − x = 0
Q. 46.
Q
x(3x − 1) = 0
1
x = − __
2
Q. 47.
Q
−x2 + 6x + 16 = 9
1
x = 0 or x = __
3
x2 − 6x − 7 = 0
(8x)2 − (1)2 = 0
(x + 1)(x − 7) = 0
1
x = ± __
8
x = −1
Q. 48.
Q
2x2 + 3x − 2 = 0
Q. 39.
(7x + 6)(x + 5) = 0
6
x = − __ or x = −5
7
(3x + 5)(2x − 1) = 0
5
1
x = − __ or x = __
3
2
x(10x − 7) = 0
x=0
2
or x = 7
x2 − 1 − 99 = 0
x2 − 100 = 0
x2 = 100
(2x − 1)(x + 2) = 0
1
x = __ or x = −2
2
Q. 38.
x=4
x2 − 6x − 16 = −9
(8x − 1)(8x + 1) = 0
Q. 37.
or
64x2 − 1 = 0
Q. 35.
Q. 36.
3x2 + 4x − 7 = 0
(3x + 7)(x − 1) = 0
7
x = − __ or x = 1
3
(x − 8)(x + 1) = 0
or x = −1
5x2 + 24x + 16 = 0
(5x + 4)(x + 4) = 0
4
x = − __ or x = −4
5
x2 − 7x − 8 = 0
x=8
100x2 = 400
x2 = 4
(12x)2 − (4)2 = 0
(12x − 4)(12x + 4) = 0
4
1
x = ± __ = ± __
3
12
Q. 30.
Q. 40.
Q
x = ±10
Q.
Q 49.
x2 − 1 − 5x − 13 = 0
x2 − 5x − 14 = 0
(x − 7)(x + 2) = 0
x=7
Q.
Q 50.
7
or x = __
10
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
or x = −2
3x2 − 13x − 30 = 0
(3x + 5)(x − 6) = 0
5
x = − __ or x = 6
3
Q. 51.
5x2 − 31x + 6 = 0
(5x − 1)(x − 6) = 0
1
x = __ or x = 6
5
Q. 52.
10x2 − 3x − 4 = 0
(5x − 4)(2x + 1) = 0
4
1
x = __ or x = − __
5
2
Q. 53.
Q
Q. 5.
=4
Q
Q. 6.
=
Q
Q. 7.
Q. 2.
Q. 3.
Q. 4.
−6 ± √ 36 − 20
________________
2
___
−6 ± √16
= __________
2
−6 ± 4
= _______
2
= −1 or −5
________
−5 ± √ 25 + 56
_______________
x=
2
___
−5 ± √81
__________
=
2
−5 ± 9
_______
=
2
= 2 or −7
________
7 ± √ 49 − 48
_____________
x=
__2
7 ± √1
= _______
2
7±1
______
=
2
= 4 or 3
________
−1 ± √ 1 + 48
_______________
x=
2
___
−1 ± √49
__________
=
2
−1 ± 7
_______
=
2
= 3 or −4
3
__
or
2
__________
________
x=
4
= −4
Exercise 7.2
Q. 1.
−5 ± √ 121
___________
−5 ± 11
= ________
4
5x2 + 7x + 2 = 0
(5x + 2)(x + 1) = 0
2
x = − __ or x = −1
5
or −9
________
−5 ± √ 25 + 96
_____________
x=
4
____
2x2 + x − 21 = 0
(2x + 7)(x − 3) = 0
7
x = − __ or x = 3
2
Q. 54.
_________
−5
± √25 + 144
________________
x=
2
_____
−5 ± √ 169
= ____________
2
−5 ± 13
________
=
2
Q
Q. 8.
−11 ± √ 121 −112
x = __________________
4
__
−11 ± √ 9
__________
=
4
−11 ± 3
________
=
4
−7
= −2 or ___
2
______
_
−2 ± √4 + 60
x = ______________
6___
−2
± √ 64
__________
=
6
−2 ± 8
______
=
6
5
= 1 or − __
3
________
Q
Q. 9.
x=
−3 ± √ 9 + 112
_______________
8____
−3 ± √121
= ___________
8
−3 ± 11
= ________
8
−7
= ___ or 1
4
________
Q. 10.
Q
x=
5 ± √ 25 + 56
_____________
14
___
5
±
81
√
= ________
14
5±9
_____
=
14
=1
2
or − __
7
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
3
___________
__________
Q. 11.
x=
26 ± √ 676 + 108
_________________
6
____
Q. 17.
Q
26 ± √ 784
= __________
6
26 ± 28
_______
=
6
1
= 9 or − __
3
x=
=
=
=
__________
Q. 12.
−15 ± √ 225 + 216
x = ___________________
_4__
−15
±
441
√
= ____________
4
−15 ± 21
= ________
4
3
__
=
or −9
2
=
x=
Q. 18.
Q
8 ± √ 64 + 260
_______________
26
8 ± √ 324
= _________
26
8 ± 18
______
=
26
5
= 1 or − __
13
x=
Q. 19.
Q
−28 ± √ 784 + 240
___________________
10
_____
−28
±
1024
√
_____________
=
10
−28 ± 32
= ________
10
2
= __ or −6
5
x=
Q 20.
Q.
=
− 16 ± √ 256 − 112
___________________
56
____
− 16 ± √ 144
____________
56
− 16 ± 12
= __________
56
1
1
__
=−
or − __
2
14
Q. 16.
4
x=
9 ± √ 81 − 80
_____________
4
___
1
±
81
√
= ________
4
1±9
= _____
4
5
__
=
or −2
2
10
9 ± √1
= _______
10
9±1
_____
=
10
4
= 1 or __
5
__
_________
Q. 21.
Q
14 ± √ 196 − 96
_________________
x=
48
____
14 ± √100
= ___________
48
14 ± 10
= ________
48
1
1
___
=
or __
2
12
x2 + x − 2 =______
0
−1 ± √ 1 + 8
x = _____________
2 __
−1 ± √ 9
= _________
2
−1 ± 3
_______
=
2
= 1 or −2
x=
1 ± √ 1 + 80
____________
6
___
19
±
25
√
= _________
6
19
±
5
= _______
6
7
= 4 or __
3
________
__________
Q. 15.
y=
19 ± √ 361 − 336
_________________
_______
____
__________
Q. 14.
28
51 ± √ 2209
____________
28
51 ± 47
_______
28
98
4
__
or __
28
28
7
1
__
or __
7
2
_____
__________
_________
Q. 13.
51 ± √ 2601 − 392
__________________
Q. 22.
Q
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
a2 + 3a + 2 ______
=0
−3 ± √ 9 − 8
a = _____________
2 __
−3 ± √ 1
= _________
2
−3 ± 1
______
=
2
= −2 or −1
Q. 23.
6x2 + 17x + 10
=0
__________
−17 ± √ 289 − 240
x = __________________
12
___
−17 ± √ 49
= ___________
12
−17 ± 7
= _______
12
5
= − __ or −2
6
________
Q. 29.
Q
Q. 30.
Q
________
−1 ± √ 1 + 528
_______________
Q. 24.
x=
Q. 25.
2x2 − 15x +_________
7=0
2____
−1
±
529
√
= ___________
2
−1 ± 23
= _______
2
= 11 or −12
15 ± √ 225 − 56
x = ________________
4
____
15 ± √ 169
= __________
4
15 ± 13
= _______
4
1
= 7 or __
2
Q. 26.
Q. 27.
Q. 28.
2x2 + 39x + 85
=0
___________
−39 ± √ 1521 − 680
x = ____________________
4
____
−39
±
841
√
= ____________
4
−39
±
29
= ________
4
5
= − __ or −17
2
36x2 − 3x = 0
12x2 − x __
=0
1 ± √1
x = _______
24
1±1
_____
=
24
1
__
=
or 0
12
5x2 − 17x −__________
12 = 0
17
±
289
+ 240
√
x = _________________
10
____
17 ± √ 529
__________
=
10
17 ± 23
_______
=
10
3
= 4 or − __
5
x=
0 ± √ 0 + 256
_____________
2
____
0
±
256
√
= _________
2
16
= ±__
2
= ±8
5
17
10x2 − __ x − __ = 0
8
16
160x2 − 34x____________
−5=0
34 ± √ 1156 + 3200
___________________
x=
320
_____
34
±
4356
√
= ____________
320
34 ± 66
= _______
320
32
100
___
=
or − ___
320
320
5
1
= __ or − __
16
10
Q. 31.
Q
2x2 − 5x =_______
0
5 ± √ 25 + 0
____________
x=
4
___
5 ± √ 25
= ________
4
5±5
_____
=
4
5
__
=
or 0
2
Q. 32.
Q
3x2 − 21x = 0
x2 − 7x _______
=0
7
±
√ 49 + 0
x = ____________
2
___
7 ± √ 49
= ________
2
7±7
_____
=
2
= 7 or 0
Q. 33.
Q
x2 + 6x + 9 ________
=0
−6 ± √ 36 − 36
x = _______________
2
−6 ± 0
______
=
2
= −3
Q. 34.
Q
30x2 + 11x −__________
2=0
−11
±
121
+ 240
√
x = __________________
60
____
−11 ± √ 361
____________
=
60
−11 ± 19
________
=
60
1
2
__
=−
or __
2
15
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
5
Exercise 7.3
Q. 1.
Q. 2.
Q. 3.
Q. 4.
__
−4 ± √ 8
_________
x=
2
= −0.6 or −3.4
___
−9 ± √ 57
__________
x=
2
= −0.7 or −8.3
___
−9 ± √ 73
__________
x=
4
= −0.11 or −4.39
____
5 ± √ 137
_________
x=
2
= 8.35 or −3.35
____
Q. 5. x =
8 ± √ 264
_________
20
= 1.21 or −0.41
___
−9 ± √ 33
__________
Q. 6. x =
12
= −0.3 or −1.2
____
3 ± √ 193
_________
Q. 7. x =
4
= 4.22 or −2.72
__
−6 ± √ 0
_________
Q. 8. x =
18
1
= −__
3
= −0.333
____
Q. 9. x =
2 ± √ 220
_________
= 1.4
12
or −1.1
_____
Q. 10. x =
16 ± √ 1768
___________
36
or −0.7
____
−5 ± √ 205
___________
Q. 11. x =
6
= 1.55 or −3.22
____
8 ± √ 512
_________
Q. 12. x =
14
= 2.2 or −1.0
___
−6 ± √ 20
__________
Q. 13. x =
2 __
−6 ± 2 √ 5
__________
=
2 __
= −3 ± √ 5
= 1.6
6
___
Q.
Q 14. x =
=
8 ± √ 28
________
2 __
8 ± 2 √7
________
2 __
= 4 ± √7
____
Q.
Q 15. x =
−2 ± √504
___________
10 ___
−2 ± 6 √ 14
= ___________
10 ___
−1 ± 3 √ 14
___________
=
5
____
1 ± √ 157
_________
Q. 16. x =
Q
6
= 2.25 or −1.92
____
−8 ± √376
___________
Q 17. b =
Q.
12
= 0.9 or −2.3
____
Q.
Q 18. x =
=
Q.
Q 19. x =
30 ± √ 720
__________
90 __
5 ± 2 √5
________
15
____
−8 ± √160
___________
16 ___
−8
±
4 √ 10
= ___________
16 ___
−2 ± √ 10
= __________
4
____
−4 ± √128
___________
Q.
Q 20. x =
14 __
−4
±
8 √2
= __________
14 __
−2 ± 4 √ 2
= __________
7
____
2 ± √ 224
_________
Q.
Q 21. x =
10 ___
2 ± 4 √ 14
_________
=
10 ___
1
±
2 √ 14
= _________
5
= 1.70 or −1.30
____
−2 ± √340
___________
Q.
Q 22. x =
8 ___
−2
±
2 √ 85
= ___________
8 ___
−1 ± √ 85
= __________
4
= 2.1 or −2.6
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
_____
−7 ± √ 1393
Q. 23. x = ____________
112
or −0.396
___
−4 ± √ 76
__________
Q. 24. x =
10 ___
−4
±
2 √ 19
= ____________
10 ___
−2 ± √ 19
= __________
5
____
−2 ± √ 564
___________
Q. 25. x =
10 ____
−2 ± 2 √ 141
_____________
=
10____
−1
±
√141
= ___________
5
___
−4 ± √ 72
__________
Q. 26. x =
14 __
−4 ± 6 √ 2
__________
=
14 __
−2 ± 3 √ 2
__________
=
7
Q
Q. 2.
x2 + 1 = 2x
= 0.271
Q. 27.
4x2
+x−6=0
___
−1 ± √ 97
__________
x=
8
= 1.106 or −1.356
Q. 28. 9x2 − 8 = 0
____
0 ± √ 288
_________
x=
18
= ± 0.9428
____
−6 ± √ 156
___________
Q. 29. x =
6 ___
−6 ± 3 √ 39
= ___________
6 ___
−3 ± √ 39
__________
=
3
____
45 ± √ 105
Q. 30. __________
48
= 1.15
or
0.72
1
x + __
x =2
x2 – 2x + 1 = 0
(x – 1)(x – 1) = 0
x=1
Q. 3.
Q
2
3x – 5 + __
x =0
3x2 – 5x + 2 = 0
Q
Q. 4.
(3x – 2)(x – 1) = 0
2
x = __ or x = 1
3
9
1 _____
__
+
=1
8+x
x
x + 8 + 9x = x(x + 8)
10x + 8 = x2 + 8x
0 = x2 – 2x – 8
(x – 4)(x + 2) = 0
x=4
Q
Q. 5.
or x = –2
1
_____
1
7
+ __ = ___
x+3
x
10
10(x) + 10(x + 3) = 7(x)(x + 3)
10x + 10x + 30 = 7x2 + 21x
20x + 30 = 7x2 + 21x
7x2 + x – 30 = 0
Q
Q. 6.
(7x + 15)(x – 2) = 0
15
x = – __ or x = 2
7
1
1
1
_____
+ _____ = __
x+4 x+1 2
LCD = 2(x + 4)(x + 1)
2(x + 4)(x + 1)
_____________
x+4
2(x + 4)(x + 1)
+ _____________
x+1
1(2)(x + 4)(x + 1)
= _______________
2
2(x + 1) + 2(x + 4) = (x + 4)(x + 1)
2x + 2 + 2x + 8 = (x2 + 5x + 4)
x2 + 5x + 4 = 4x + 10
Exercise 7.4
Q. 1.
5
__
x2 – x – 6 = 0
(x + 3)(x – 2) = 0
2
__
x + x =7
5
2
__
x __
x + x x = x(7)
5 + 2 = 7x
( ) ( )
x = –3
or x = 2
7 = 7x
1=x
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
7
Q. 7.
2
_____
x+1
4
+ ______ = 2
2x – 1
LCD = 2(x + 1)(2x – 1)
3
3
4
Q
Q. 11. ______ − _______ = −__
2x − 5 5x − 1
1
4(5x − 1) − 3(2x − 5) = −3(2x − 5)(5x − 1)
4(x + 1)(2x – 1)
2(x + 1)(2x – 1)
___________
+ ______________
x+1
2x – 1
20x − 4 − 6x + 15 = −3(10x2 − 27x + 5)
14x + 11 = −30x2 + 81x − 15
= 2(x + 1)(2x – 1)
30x2 − 67x + 26 = 0
2(2x – 1) + 4(x + 1) = 2(x + 1)(2x – 1)
(15x − 26)(2x − 1) = 0
26
1
x = ___ or x = __
2
15
4
1
______
___
Q
Q. 12.
=5−
2x − 3
3x
30x2 − 56x + 12 = 0
4x – 2 + 4x + 4 = 2(2x2 + x – 1)
8x + 2 = 4x2 + 2x – 2
4x2 – 6x – 4 = 0
2x2 – 3x – 2 = 0
15x2 − 28x + 6 = 0
(2x + 1)(x – 2) = 0
2x + 1 = 0
2x = –1
1
x = – __
2
Q. 8.
a = 15 b =
−28 c = 6
________________
2 − 4(15)(6)
28
±
(−28)
√
x = ______________________
2(15)
or x – 2 = 0
or
____
x=2
=
5
_____
3
– ______ = 1
x + 1 3x – 1
15x – 5 – 3x – 3 = 3x2 + 2x – 1
1
1
Q
Q. 13. _____ + _____ = 2
x+3 x+4
2x2 + 12x + 17 = 0
12x – 8 = 3x2 + 2x – 1
x=
3x2 – 10x + 7 = 0
x=1
10(6x – 1)(2x + 1) – 5(2x + 1) = 15(6x – 1)
2
2(2)
x = −2.29, −3.71
2
4
Q
Q. 14. _____ + _____ = 1
x−3 x+2
2=0
x2 − 7x + ______________
10(12x2 + 4x – 1) – 10x – 5 = 90x – 15
x=
120x2 + 30x – 15 = 90x – 15
120x2 – 60x = 0
2x2 – x = 0
45x + 75 = 11x2 + 44x + 33
___
=
7 ± √ 41
________
84 ± √ (−84)2 − 4(70)(17)
x = _______________________
2(70)
1
or x = __
2
15x + 45 + 30x + 30 = 11x2 + 44x + 33
2
3
5
Q
Q. 15. ______ − ______ = 5
2x − 1 7x − 2
70x2 − 84x _________________
+ 17 = 0
x(2x – 1) = 0
15(x + 3) + 2(15)(x + 1) = 11(x + 1)(x + 3)
√
7 ± (−7) − 4(1)(2)
___________________
2(1)
x = 6.70, 0.30
120x2 + 40x – 10 – 10x – 5 = 90x – 15
____
=
84 ± 2√ 574
___________
140
x ≈ 0.94, 0.26
4
1
1
Q
Q. 16. _____ − ______ = __
x − 5 2x + 4 3
2x2 − 27x −_________________
83 = 0
11x2 − x − 42 = 0
(11x + 21)(x − 2) = 0
21
x = −__ or x = 2
11
8
______________
−12 ± 2√2
= ___________
4
7
or x = __
3
5
15
10 – ______ = ______
6x – 1 2x + 1
x=0
1
2
11
_____
_____
___
+
=
x + 1 x + 3 15
√
−12 ± (12) − 4(2)(17)
______________________
__
(x – 1)(3x – 7) = 0
Q. 10.
30
x = 1.62, 0.25
5(3x – 1) – 3(x + 1) = 1(x + 1)(3x – 1)
Q. 9.
28 ± 2√ 106
___________
27 ± √ (−27)2 − 4(2)(−83)
x = ________________________
2(2)
______
27 ± √ 1,393
= ____________
4
x = 16.08, −2.58
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
ti
2
Exercise 7.5
Q
Q. 14. x(x + 6) = 0
x2 + 6x = 0
Q. 1. (x − 3)(x − 4) = 0
∴b=6
x2 − 7x + 12 = 0
c=0
Q. 2. (x − 2)(x − 5) = 0
Q.
Q 15. x = p
(x − p)(x − p) = 0
x2 − 7x + 10 = 0
Q. 3. (x + 1)(x − 2) = 0
x2 − x − 2 = 0
Q. 4. (x +
3)2
x=p
x2
− px − px + p2 = 0
x2 − 2px + p2 = 0
−2px = 8x
=0
−2p = 8
p = −4
x2 + 6x + 9 = 0
∴ r = (−4)2 = 16
Q. 5. (x − 11)(x + 1) = 0
x2 − 10x − 11 = 0
Q. 6. (x − 7)(x − 0) = 0
x2 − 7x = 0
Q. 16. Call root a
Q
(x − a)2 = 0
x2 − 2ax + a2 = 0
∴ −2a = −12
Q. 7. (x + 8)(x − 8) = 0
x2
− 64 = 0
Q. 8. (x − 0)(x + 4) = 0
x2 + 4x = 0
Q. 9. (x + 3)(x − 3) = 0
x2 − 9 = 0
Q. 10. x = p
x = 2p
(x − p)(x − 2p) = 0
x2 − 2px − px + 2p2 = 0
x2 − 3px + 2p2 = 0
Q. 11. x = p
x=q
a=6
∴c=
a2
= 62 = 36
Exercise 7.6
E
Q
Q. 1. x = 5 − y
x2 + y2 = 17
(5 − y)2 + y2 = 17
25 − 10y + y2 + y2 = 17
2y2 − 10y + 8 = 0
y2 − 5y + 4 = 0
(y − 1)(y − 4) = 0
y=1
or y = 4
(x − p)(x − q) = 0
x=4
or x = 1
x2 − xq − px + pq = 0
(4,1)
x2
− (p + q)x + pq = 0
Q. 12. x = p
x = −p
or
(1,4)
Q. 2. x = 1 + y
Q
x2 + y2 = 25
(x − p)(x + p) = 0
(1 + y)2 + y2 = 25
x2 + px − px − p2 = 0
1 + 2y + y2 + y2 = 25
x2 − p2 = 0
2y2 + 2y − 24 = 0
Q. 13. (x − 5)(x − 3) = 0
x2 − 8x + 15 = 0
∴ b = −8
c = 15
y2 + y − 12 = 0
(y + 4)(y − 3) = 0
y = −4
or y = 3
x = −3
or x = 4
(−3,−4)
or
(4,3)
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
9
3
x = − __
5
Q. 3. x + y = 4
3
3
y = 4 − __ + 3 = __
5
5
x2 + y2 = 16
x2 + (4 − x)2 = 16
(
33
∴ − __,__
55
x2 + 16 − 8x + x2 = 16
2x2 − 8x = 0
)
or
x2 + y2 = 2
x=0
x2 + x2 = 2
or x = 4
y = 4 or y = 0
or
(4,0)
(−1,−1)
x2 = 1
x = ±1
⇒x=6+y
y = ±1
x2 + y2 = 26
∴ (1,1)
(6 +
+
y2
= 26
36 + 12y + y2 + y2 = 26
x2 + y2 = 13
x2 + (5x + 13)2 = 13
(y + 1)(y + 5) = 0
or y = −5
x=5
or x = 1
(5,−1)
or
x2 + 25x2 + 130x + 169 = 13
26x2 + 130x + 156 = 0
2x2 + 10x + 12 = 0
(1,−5)
x2 + 5x + 6 = 0
Q. 5. y = −x − 2 = (−1)(x + 2)
(x + 2)(x + 3) = 0
x2 − y2 = 60
x = −2
x2 − (x + 2)2 = 60
or x = −3
y = 5(−2) + 13 or y = 5(−3) + 13
x2 − (x2 + 4x + 4) = 60
=3
−4x − 4 = 60
= −2
∴ (−2,3)
−4x = 64
x = −16
(−1,−1)
⇒ 5x + 13 = y
y2 + 6y + 5 = 0
y = −1
or
Q.
Q 8. 5x − y = −13
2y2 + 12y + 10 = 0
or
(−3,−2)
Q.
Q 9. x − 2y = −5
y = (−1)(−16 + 2)
⇒ x = 2y − 5
y = 14
x2 + y2 = 10
∴ (−16,14)
Q. 6. 4x − y = −3
(2y − 5)2 + y2 = 10
4y2 − 20y + 25 + y2 − 10 = 0
⇒ 4x + 3 = y
5y2 − 20y + 15 = 0
x2 = y2
y2 − 4y + 3 = 0
x2 = (4x + 3)2
(y − 3)(y − 1) = 0
x2
+ 24x + 9
y=3
or y = 1
0 = 15x2 + 24x + 9
x=1
or x = −3
5x2
(1,3)
=
16x2
= −1
2x2 = 2
Q. 4. x − y = 6
y)2
or y = 9(−1) + 3
Q. 7. x = y
Q
2x(x − 4) = 0
(0,4)
x = −1
( )
⇒y=4−x
+ 8x + 3 = 0
(5x + 3)(x + 1) = 0
10
or
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
or
(−3,1)
Q. 10. x + 2y = 7
Q 14. x − y = 4
Q.
⇒ x = 7 − 2y
⇒x=4+y
x2
y2 + 7x = 18
+
3y2
= 28
(7 − 2y)2 + 3y2 = 28
y2 + 7 (4 + y) = 18
49 − 28y + 4y2 + 3y2 = 28
y2 + 7y + 10 = 0
7y2 − 28y + 21 = 0
(y + 2)(y + 5) = 0
y2 − 4y + 3 = 0
(y − 1)(y − 3) = 0
y=1
or y = 3
x=5
or x = 1
(5,1)
or
(1,3)
y = −2
or y = −5
x=2
or x = −1
(2,−2)
or
(−1,−5)
Q
Q. 15. x − 2y = 1
⇒ x = 1 + 2y
Q. 11. x − 2y = −5
⇒ x = 2y − 5
x2 − 3y2 = 6
x2 + y2 = 85
(1 + 2y)2 − 3y2 = 6
(2y − 5)2 + y2 = 85
1 + 4y + 4y2 − 3y2 − 6 = 0
4y2 − 20y + 75 + y2 = 85
y2 + 4y − 5 = 0
5y2 − 20y − 60 = 0
(y + 5)(y − 1) = 0
y2 − 4y − 12 = 0
y = −5
or y = 1
(y − 6)(y + 2) = 0
x = −9
or x = 3
y=6
or y = −2
(−9,−5)
or
x=7
or x = −9
(7,6)
or
(−9,−2)
(3,1)
Q
Q. 16. x + y = 7
⇒y=7−x
Q. 12. x = y + 4
xy = 12
3x + y2 = 16
3(y + 4) + y2 = 16
x(7 − x) = 12
y2 + 3y − 4 = 0
7x − x2 = 12
(y + 4)(y − 1) = 0
0 = x2 − 7x + 12
y = −4
or y = 1
(x − 3)(x − 4) = 0
x=0
or x = 5
x=3
or x = 4
(0,−4)
or
y=4
or y = 3
(3,4)
or
(5,1)
Q. 13. x = y
x2 + 2y = 35
x2
(4,3)
Q
Q. 17. x + 2y = 11
+ 2x − 35 = 0
(x + 7)(x − 5) = 0
x = −7
or x = 5
y = −7
or y = 5
(−7,−7)
or
(5,5)
⇒ x = 11 − 2y
xy = 14
(11 − 2y)y = 14
11y − 2y2 = 14
0 = 2y2 − 11y + 14
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
11
(2y − 7)(y − 2) = 0
y=
7
__
Exercise 7.7
E
or y = 2
2
x=4
or x = 7
( 4,__72 )
or
Q. 1. Six possible roots are given for each
Q
equation. Use trial and error to determine
the correct roots.
(7,2)
(i)
x
x2 + 5x + 4
=y
⇒ x = 2y − 1
1
(1)2 + 5(1) + 4
= 10
x2 − xy = 10
4
(4)2 + 5(4) + 4
= 40
(2y − 1)2 − y(2y − 1) = 10
–1
(–1)2 + 5(–1) + 4
=0
4y2 − 4y + 1 − 2y2 + y − 10 = 0
2
(2)2 + 5(2) + 4
= 18
2y2 − 3y − 9 = 0
–2
(–2)2 + 5(–2) + 4
= –2
(2y + 3)(y − 3) = 0
3
y = − __ or y = 3
2
x = −4 or x = 5
3
−4,− __ or (5,3)
2
–4
(–4)2 + 5(–4) + 4
=0
Q. 18. x − 2y + 1 = 0
Answers: x = −1
(ii)
)
(
Q. 19. y = −2x
x
x2 – 12x + 32
=y
4
(4)2 – 12(4) + 32
=0
–4
(–4)2
– 12(–4) + 32
= 96
2x2 − y2 + xy = −16
–2
(–2)2 – 12(–2) + 32
= 60
2x2 − (−2x)2 + x(−2x) = −16
16
(16)2 – 12(16) + 32
= 96
2x2
8
(8)2 – 12(8) + 32
=0
2
(2)2
= 12
−
4x2
−
2x2
= −16
−4x2 = −16
x2
=4
(iii)
x = 2 x = −2
⇒ y = −4
(2,−4)
– 12(2) + 32
Answers: x = 4
x = ±2
or
⇒y=4
(−2,4)
Q. 20. 2x = y + 3
⇒ y = 2x − 3
x2 + y2 − 3xy = −45
x2 + (2x − 3)2 − 3x(2x − 3) = −45
x2 + 4x2 − 12x + 9 − 6x2 + 9x + 45 = 0
−x2 − 3x + 54 = 0
12
or x = −4
x
x2 – x – 12
(0)2 – (0) – 12
= –12
–1
(–1)2 – (–1) – 12
= –10
3
(3)2 – (3) – 12
= –6
–3
(–3)2 – (–3) – 12
=0
4
(4)2 – (4) – 12
=0
2
(2)2 – (2) – 12
= –10
Answers: x = −3
Q. 2.
Q
(i) x = −1
(ii) x = −3
(x + 9)(x − 6) = 0
(iii) x = −3
x = −9
or x = 6
(iv) x = −5
y = −21
or y = 9
∴ (−9,−21)
or
Q
Q. 3. 3.7
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
=y
0
x2 + 3x − 54 = 0
(6,9)
or x = 8
or −5
or x = 4
Exercise 7.8
Q.
Q 6.
y
8
Q. 1.
(i) x = −2
or
(ii) x = −2.7
(iii) x = −5
6
x=3
4
or x = −2
or
(iv) x = −0.5
2
x = −1.5
0 0.5
or x = 4.5
1
1.5
2
2.5
4
4.5
5
5.5
6
(i) (−1,−2) and (1,2)]
x ≈ 1.3
(iii) (−4,3) and (4,−3)
or x ≈ 4.5
Q 7.
Q.
6
(iv) (−1.2,0.3) and (1.6,5.7)
2
0
x
–3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 3
–2
–4
(i) Blue = C
–6
Red = B
–8
Green = A
x = −2
(ii) x2 − 9 = 0: x = ±3
x2 + x − 6 = 0: x = −3
−x2
12
or x = 2
− 4x + 21 = 0: x = −7 or x = 3
or
x = 1.5
Q. 8.
Q
y
10
8
y
6
10
4
8
2
0
6
x
–1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
–2
4
2
–4
x
0
0 0.5
–0.5
–2
1
1.5
2
2.5
3
3.5
4
(i) The curve does not cut the x-axis.
(ii) You end up getting the root of a negative
number, which is imaginary, ∴ no real roots.
See where the curve cuts the x−axis.
[At x = 1
y
4
In each case the two functions concerned
have the same x-value and the same y-value
at the point of intersection.
Q. 4.
3.5
–4
(ii) (−1.5,−2.5) and (2.5,1.5)
Q. 3.
3
–2
(v) x = −3
Q. 2.
x
0
Q. 9.
Q
or x ≈ 2.65]
Q. 5.
(i) and (iii)
y
y
6
(5,6)
6
4
4
2
0
2
x
0
0 1
–3 –2 –1
2
3
4
5
6
7
–3
–2
x
0
–1
1
2
3
4
5
6
–2
8
–2
–4 (0,–4)
–4
–6
–6
(ii) x = −1
x = −1
or
x=4
or
x=4
(iv) x = 2
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
13
(v) At x = 0, both functions take the
value −4.
Q 6.
Q.
x2 = (7x)(8)
x2 − 56x = 0
At x = 5, both functions take the value 6.
x(x − 56) = 0
x=0
∴ x = 56
Exercise 7.9
x2 + 3x = 18
Q. 1.
x2
(Answer of 0 will not satisfy original property)
Q. 7. x(2x − 5) = 7
Q
+ 3x − 18 = 0
(x + 6)(x − 3) = 0
x = −6
or x = 3
Q. 2. 5x2 + 4x = 28
5x2 + 4x − 28 = 0
(5x + 14)(x − 2) = 0
14
x = − ___ or x = 2
5
Q. 3. x − 15; − x − 3
x2 + (x − 3)2 = 65
x2 + x2 − 6x + 9 − 65 = 0
2x2 − 5x − 7 = 0
(2x − 7)(x + 1) = 0
7
x = __ or x = −1
2
x 3332
Q
Q. 8. ______
3332
_____
x
3332
∴ (x + 12) _____
x + 6 = 3332 + 1068
39,984
3332 + 6x + _______
+ 72 = 4400
x
39,984
6x + _______
− 996 = 0
x
6x2 − 996x + 39,984 = 0
(
x2 − 3x − 28 = 0
(x − 7)(x + 4) = 0
or x = −4
3332
_____
3332
_____
x = 34
x = 49
∴ Ans: 98 m, 34 m or 68 m, 49 m
∴ (7,4) and (−4,−7)
Q. 4.
x(x + 1) = 56
x2 + x − 56 = 0
Q
Q. 9. 2n(2n + 2) = 168
4n2 + 4n − 168 = 0
(x + 8)(x − 7) = 0
x = −8
n2 + n − 42 = 0
or x = 7
(n + 7)(n − 6) = 0
x > 0 since x ∈ N
n = −7
∴x=7
or n = 6
n ≠ −7 since n ∈ N
Ans: 7, 8
∴n=6
∴ Smaller number is 2(6) = 12
Q. 5. (x + 5)(x) = 234
x2 + 5x − 234 = 0
Q.
Q 10. 3n(3n + 2) = 483
(x + 18)(x − 13) = 0
9n2 + 6n − 483 = 0
x = −18
or x = 13
3n2 + 2n − 161 = 0
x > 0 as x is a distance
(3n + 23)(n − 7) = 0
23
n = – ___ or n = 7
3
since n ∈ N
∴ x = 13
∴ Ans: 13 m, 18 m
∴ 3(7) = 21
Ans: 21, 23
14
)
x2 − 166x + ___
6664 = 0
166 ± √900 166 ± 30
x = ____________ = _________ = 98 or 68
2
2
∴ x = 98 or x = 68
2x2 − 6x − 56 = 0
x=7
or x = 56
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
Q. 11.
x
x(x + 4) = 2(x + 2) + 11
Q.
Q 14.
x
x
x
12
x2 + 4x = 2x + 15
x2 + 2x − 15 = 0
10
(x + 5)(x − 3) = 0
x
x = −5
x
x
x
on x > 0, since x ∈ N
4x2 + 2(10x) + 2(12x) = 48
4x2 + 44x − 48 = 0
x2
or x = 3
∴ Ans: 3, 5, 7
Q
Q. 15.
40
+ 11x − 12 = 0
40 – 2x
(x + 12)(x − 1) = 0
x = −12;
or x = 1
30
30 – 2x
since x > 0
as x is a distance
∴ Ans = 1 m
2(40 − 2x)(30 − 2x) = 30(40)
OR
2(2)(2)(20 − x)(15 − x) = 1200
12 + 2x
(20 − x)(15 − x) = 150
300 − 35x + x2 = 150
10 + 2x
48 + 120
= 168
x2 − 35x + 150 = 0
(x − 5)(x − 30) = 0
x=5
or x = 30
(10 + 2x)(12 + 2x) = 168
Reject x = 30 since 30 − 2(30) < 0
4x2 + 44x + 120 − 168 = 0
∴x=5
4x2 + 44x − 48 = 0
Answer: 5 cm
Q. 12. x + x + 2 = (x + 1)2
2x + 2 = x2 + 2x + 1
0 = x2 − 1
x2 = 1
x = ±1
Ans: −1, 0, 1
or
1, 2, 3
(x − 4)(x) = 221
Q. 13.
x2
− 4x − 221 = 0
(x − 17)(x + 13) = 0
x = 17
or x = −13
5
Q. 16. x2 + 14 = __(x + 14)
Q
2
2x2 + 28 = 5x + 70
2x2 − 5x − 42 = 0
(2x + 7)(x − 6) = 0
7
x = − __ ∴ x = 6
2
Reject since
x>0
Girl: 6 years old
Father: 36 years old
Q
Q. 17. Amy: x + 5
Bridget: x
as x > 0, x is a distance
Caroline: 2(x + 5)
∴ Length: 17 − 4 = 13 m
2(x + 5)2 = 722
Breadth: 17 m
(x + 5)2 = 361
____
∴ x + 5 = ± √ 361
Active M
Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
15
x + 5 = ±19
∴ x = 14
Q.
Q 21.
or x = −24
since x > 0
∴ Amy: 19
Bridget: 14
Caroline: 38
Q. 18. (8x + 5)2 = (x + 4)2 + (7x + 5)2
50x2
+ 80x + 25 =
€400
No of
members
x
x+2
Prize per
member
400
___
400
____
x
x+2
400x + 800 − 400x = 10x2 + 20x
14x2 + 2x − 16 = 0
10x2 + 20x − 800 = 0
7x2 + x − 8 = 0
x2 + 2x − 80 = 0
(7x + 8)(x − 1) = 0
8
x = − __ or x = 1
7
8
since 7 − __ + 5
7
(x + 10)(x − 8) = 0
x = −10,
1
Q.
Q 22. x + __
x = 2.5
x2 + 1 = 2.5x
∴ x = 1 ∴ lengths: 8(1) + 5 = 13
x2 − 2.5x + 1 = 0
1+4=5
2x2 − 5x + 2 = 0
7(1) + 5 = 12
(2x − 1)(x − 2) = 0
1
x = __ or x = 2
2
Answer: x = 2
Q. 19. 22(7 − 2x) = 110
7 − 2x = 5
2 = 2x
Q.
Q 23.
x=1
∴ Width = 7 − 2
5n2 + 10n − 600 = 0
∴ Volume = 22(5)(1)
n2 + 2n − 120 = 0
= 110 cm3
(ii)
A to B
B to C
Distance (km)
10
12
Speed (km/hr)
x
x–1
Time (hrs)
10
__
12
____
10
___
x
n + 12
n – 10
(n + 12)(n − 10) = 0
n + 12 = 0
x–1
n = −12
12
_____
+
=5
x
x−1
10(x − 1) + 12(x) = 5(x)(x − 1)
10x − 10 + 12x =
5x2
300
300 _____
___
n −n+2=5
300(n + 2) − 300n = 5(n)(n + 2)
300n + 600 − 300n = 5n2 + 10n
= 5 cm
(i)
or x = 8
Answer: x = 8
= −3 < 0
5x2
−5
or n − 10 = 0
or
n = 10
Answer: 10 winners
15
10
Q. 24. ___ = _____
Q
x+2
x
− 27x + 10 = 0
(5x − 2)(x − 5) = 0
2
x = __ or x = 5
5
3
2
__
− 1 = −__
5
5
2
∴ x = __
∴x=5
5
Answer: x = 5
16
€400
400(x + 2) − 400(x) = 10(x)(x + 2)
+ 78x + 41
( )
Q. 20.
Total Prize
400
400
(ii) ____ − _____ = 10
x+2
x
64x2 + 80x + 25 = x2 + 8x + 16 + 49x2
+ 70x + 25
64x2
(i)
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
10(x + 2) = 15(x)
10x + 20 = 15x
20 = 5x
4=x
(i) ∴ 1st part: 4 km/hr, 2nd part: 6 km/hr
10 = 2.5 hrs or 2 hrs 30 mins.
(ii) ∴ ___
4
20
20
Q. 25. Monday: __
Tuesday: ____
x−2
x
20 5
20
_____
− ___ = __
x−2
6
x
5
__
So 50 mins = hr
6
20(6)(x) − 20(6)(x − 2) = 5(x)(x − 2)
120x − 120x + 240 =
5x2
5x2
− 10x
− 10x − 240 = 0
x2
(x + 6)(x − 8) = 0
or x = 8
∴ Monday: 8 km/hr,
}
Tuesday: 6 km/hr
45
Q. 26. Ben: __
x
45 ____
45
1
__
−
= __
45
____
x
x+1 2
Ann:
x+1
2(x + y) = 2.4
x(1.2 − x) = 0.35
x + y = 1.2
1.2x −
x2
= 0.35
y = 1.2 − x
x2 − 1.2x + 0.35 = 0
10x2 − 12x + 35 = 0
100x2 − 120x + 35 = 0
20x2 − 24x + 7 = 0
− 2x − 48 = 0
x = −6
xy = 0.35
Q
Q. 28.
(10x − 7)(2x − 1) = 0
7
1
x = ___ or x = __
2
10
x = 0.7 or x = 0.5
y = 0.5
or y = 0.7
∴ Length = 0.7 m; Width = 0.5 m
Q. 29. (n + 1)3 − n3 = 61
Q
2(45x + 45 − 45x) = x2 + x
(n + 1)(n + 1)(n + 1) − n3 = 61
90 = x2 + x
n3 + 3n2 + 3n + 1 − n3 = 61
x2 + x − 90 = 0
3n2 + 3n − 60 = 0
(x + 10)(x − 9) = 0
n2 + n − 20 = 0
x=9
(n + 5)(n − 4) = 0
∴ Ben: 9 km/hr
n = −5
Ann: 10 km/hr
45
45
(Check: ___ = 5; ___ = 4.5;
9
10
1
5−4.5 = 0.5 = __)
2
as n > 0, n ∈ N
Q. 30. 2(x + y) = 32 ⇒ x + y = 16 ⇒ y = 16 − x
Q
⇒ −x2 + 16x − 60 = 0
xy = 18
18
∴ y = ___
x
18
∴ 4x + 1 = ___
x
x2 − 16x + 60 = 0
(x − 10)(x − 6) = 0
∴ 4x2 + x − 18 = 0
(4x + 9)(x − 2) = 0
9
x = − __ or x = 2
4
y = −8 or y = 9
)
9
∴ − __,−8 or (2,9)
4
9
But − __,−8 will not satisfy the
4
second condition
(
∴ Answer is: 4, 5
xy = 60 ⇒ x(16 − x) = 60
Q. 27. 4x + 1 = y
(
or n = 4
)
x = 10
y=6
or x = 6
or
y = 10
∴ Length = 10 m
Q
Q. 31. Originally:
Now:
l = 4x
= 4x
b = 2x
= 4x
1
__
x
2
h=x
(i) 0%
Width = 6 m
(ii) 0%
∴ x = 2, y = 9
OR
−
9
__
4
is not natural
(
9
∴ reject − __,−8
4
)
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
17
Revision Exercises
Q. 32. Distance = Speed × Time
(
(
)
)
15
Monday: x = 40 y + __
60
50
Tuesday: x = 60 y − ___
60
(
) (
15
50
40 y + ___ = 60 y − ___
60
60
40y + 10 = 60y − 50
Q. 1. (a)
(ii) x = 2
)
)
(ii) −2
(c)
(i) x2 + (x + 1)2 = 61
2x2 + 2x − 60 = 0
Q. 33. x = − 3 − y = (−1)(3 + y)
x2 + x − 30 = 0
(−1)2(3 + y)2 + y2 = 29
9 + 6y + y2 + y2 − 29 = 0
(x + 6)(x − 5) = 0
x = −6 ∴ x = 5 (since x ∈ N)
+ 6y − 20 = 0
y2 + 3y − 10 = 0
Ans: 5, 6
(y + 5)(y − 2) = 0
(ii) x − y = 3
or y = 2
xy = 28
x=3+y
or x = −5
(3 + y)(y) = 28
y2
∴ (2, −5)
(ii)
2 km East, 5 km South
y = −7
or y = 4
5 km West, 2 km North
x = −4
or x = 7
√
or
__________
22
+
(−5)2
(−5, 2)
+ 3y − 28 = 0
(i)
(iii)
(y + 7)(y − 4) = 0
_______
= √___
4 + 25
= √ 29 km (each)
Q. 2. (a) (i) x = −1 or x = 6
5
4
(ii) x = − __ or x = −__
2
3
1
(iii) x = −4 or x = __
2
3
(iv) x = __ or x = 3
5
3
7
(v) x = − __ or x = __
5
2
y
(b) (i)
20
15
10
5
x
0
0 1
–3 –2 –1
2
3
4
5
6
7
–5
18
or −5
(iii) −2
(ii) Time = 3 hrs
x=2
or x = 2
3
(v) x = −4 or x = __
2
10
(vi) x = ± ___
7
(b) (i) −3 or −5
3=y
y = −5
or x = 10
(iv) x = ±10
15
x = 40 3 + ___ = 130
60
∴ (i) Distance = 130 km
2y2
or x = 4
(iii) x = −3
60 = 20y
(
(i) x = −2
x = ±2
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
(ii)
y
8
6
4
2
0
–4
–3
–2
x
0
–1
1
2
3
4
5
6
7
0.5
1
1.5
–2
–4
–6
–8
x = −2
or x = 4
(iii)
y
90
80
70
60
50
40
30
20
10
0
–3.5 –3 –2.5 –2 –1.5 –1 –0.5
x = −2
x
0
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
or x ≈ 0.3
(i) x2 − 4 = 0
(ii)
x2 = 4
x2 − 2x − 8 = 0
(iii)
3x2 + 5x − 2 = 0
(3x − 1)(x + 2) = 0
1
x = ±2
x = 4 or x = −2
x = __ or x = −2
3
Algebra: No problem/inaccuracy in reading roots from graph is involved.
(c)
(i)
(x − 4)(x + 2) = 0
(y + 1)2 + y2 = 13
2y2
(iii) y = 3 − x
+ 2y − 12 = 0
x2 + (3 − x)2 = 5
y2 + y − 6 = 0
x2 + 9 − 6x + x2 − 5 = 0
(y + 3)(y − 2) = 0
2x2 − 6x + 4 = 0
y=2
or y = −3
x2 − 3x + 2 = 0
x=3
or x = −2
(x − 1)(x − 2) = 0
∴ (−2,−3)
or
(2,3)
x=1
or x = 2
y=2
or y = 1
∴ (1,2)
(ii)
x2
+ 2(x + 2) = 12
or
(iv) x = 4 + y
x2 + 2x − 8 = 0
y2 + 7(4 + y) = 18
(x + 4)(x − 2) = 0
y2 + 7y + 10 = 0
x = −4
(y + 2)(y + 5) = 0
or x = 2
(2,1)
y = −2 or y = 4
y = −2
∴ (−4,−2) or (2,4)
x=2
or y = −5
or x = −1
∴ (2,−2)
or
(−1,−5)
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
19
___
Q. 3. (a)
(i) x =
−2 ± √ 24
__________
Using quadratic formula:
___
2
= 1.45 or −3.45
(i) x =
2
= 2.3 or ___−1.3
−2 ± √ 40
(ii) x = __________
6
= 0.7 or
−1.4
___
____
(ii) x =
12 ± √ 124
__________
2
= 11.57 or
0.43
___
−1 ± √ 41
(iii) x = __________
2
= 2.70 or −3.70
2 ± √ 44
(iii) x = ________
4
= 2.2 or −1.2
___
(iv) x =
3 ± √ 89
________
(c) (i) x2 = 5x + 14
2
= 6.22 or −3.22
x2 − 5x − 14 = 0
___
(x − 7)(x + 2) = 0
−2 ± √ 12
(v) x = __________
4
= 0.37 or −1.37
(b) (i)
10
x = −2
or x = 7
(ii) x2 = 7x − 10
x2 − 7x + 10 = 0
y
(x − 2)(x − 5) = 0
8
x=2
6
Q. 4. (a)
4
or
x=5
(i) (x − 2)(x − 6) = 0
x2 − 8x + 12 = 0
2
x
0
–4 –3 –2 –1
1 ± √ 13
________
0 1
2
3
4
(ii) (x − 1)(x − 5) = 0
5
x2 − 6x + 5 = 0
–2
(iii) (x − 11)(x + 2) = 0
x ≈ −1.3
x2 − 9x − 22 = 0
or x ≈ 2.3
(ii)
(iv) (x + 10)(x − 4) = 0
y
x2 + 6x − 40 = 0
25
(v)
20
(x + 5)(x + 7) = 0
x2 + 12x + 35 = 0
15
(b) (i) x2 − 1 − 24 = 0
10
x2 = 25
5
0 1
–4 –3 –2 –1
x ≈ −1.4
2
3
4
(ii) 2x2 − x − 3x = 0
5
2x2 − 4x = 0
or x ≈ 0.7
(iii)
x = ±5
x
0
x2 − 2x = 0
x(x − 2) = 0
y
x=0
15
or x = 2
(iii) 21x2 − x2 + 7x − 6 = 0
10
20x2 + 7x − 6 = 0
5
x
0
–4
–3 –2 –1
0
1
2
3
4
5
–5
x ≈ −1.2
20
or x ≈ 2.2
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
(5x − 2)(4x + 3) = 0
3
2
x = __ or x = − __
5
4
(iv) 9x2 = 12x − 4
9x2 − 12x + 4 = 0
(3x − 2)(3x − 2) = 0
2
x = __
3
(v) 16 − 2x + 8x − x2 = 0
(b)
(i) 2x2 = (x + 1)2 + 34
x2 − 6x − 16 = 0
2x2 = x2 + 2x + 35
(x − 8)(x + 2) = 0
x2 − 2x − 35 = 0
x=8
(x − 7)(x + 5) = 0
or x = −2
x=7
(vi) 10x2 + 35x − 12x − 42 = 0
since x ∈ N
10x2 + 23x − 42 = 0
∴ 7, 8
(5x − 7)(x + 6) = 0
7
x = __
5
(ii) 3x2 = x(x + 2) + 60
or x = −6
3x2 = x2 + 2x + 60
(c) (i) x = 3 + y
(3 +
y)2
2x2 − 2x − 60 = 0
+ y = 17
x2 − x − 30 = 0
9 + 6y + y2 + y − 17 = 0
(x − 6)(x + 5) = 0
y2 + 7y − 8 = 0
x=6
(y + 8)(y − 1) = 0
y = −8
or y = 1
x = −5
or x = 4
∴ (−5, −8)
or
(4, 1)
(ii) x = 2y
(2y)2 + y2 = 5
y = ±1
x = ±2
(−2, −1)
(iii) x = 1 − y
(1 − y)2 + y2 = 1
1 − 2y + y2 + y2 = 1
2y2 − 2y = 0
y2 − y = 0
y(y − 1) = 0
y=0
or y = 1
x=1
or x = 0
∴ (1,0)
or
56
(i) x = 1 + ___
x
⇒ x2 = x + 56
x2 − x − 56 = 0
x – 8
+
7
x−8=0
y2 = 1
or
Q
Q. 6. (a)
(x − 8)(x + 7) = 0
5y2 = 5
∴ (2, 1)
or x = −5
since x ∈ N ∴ 6, 8
x
4y2 + y2 = 5
or x = −5
(0,1)
Q. 5. (a) (i) x2 – 3x + 1 = 0
(ii) 2x2 – 12x + 1 = 0
(iii) 3x2 − 12x + 7 = 0
(iv) 5x2 − 14x + 4 = 0
x=8
2
__
or x + 7 = 0
or
x = −7
(ii) 7 = 5x + x
⇒ 7x = 5x2 + 2
5x2 − 7x + 2 = 0
5x – 2
x – 1
(5x − 2)(x − 1) = 0
5x − 2 = 0
or x − 1 = 0
5x = 2
or
2
x = __
5
3
2
2
_____
(iii)
+ _____ = __
x−3 x−1 1
x=1
⇒ 3(x − 1) + 2(x − 3) = 2(x − 3)(x − 1)
3x − 3 + 2x − 6 = 2(x2 − 4x + 3)
5x − 9 = 2x2 − 8x + 6
2x2 − 13x + 15 = 0
2x – 3
(v) 7x2 + 4x − 2
x
–
5
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
21
(2x − 3)(x − 5) = 0
(v) Graph the function y = x2 − 3.
2x − 3 = 0
Note where y = x2 − 3 intersects
y = 2x2 − 7x + 3.
2x = 3 or x − 5 = 0
3
x=5
x = __ or
2
18
(b) ___
x =x+3
⇒ 18 = x2 + 3x
The x-values of the points of
intersection are the required
solutions.
(b)
x2 + 3x − 18 = 0
x + 6
x2 + 8x + 16 = 0
p=8
x2 − 20x + 100 = 0
b = −20
or x − 3 = 0
x = −6
or
or q = 16
(x − 10)2 = 0
(ii)
x – 3
(x + 6)(x − 3) = 0
x+6=0
(x + 4)2 = 0
(i)
or c = 100
(iii) Call root a.
x=3
(x − a)2 = 0
3
5
1
(c) _____ + _____ = __
x+1 x−1 4
⇒ 3(4)(x − 1) + 1(4)(x + 1) = 5(x + 1)(x − 1)
x2 − 2ax + a2 = 0
−2a = −10
12x − 12 + 4x + 4 = 5(x2 − 1)
a=5
16x − 8 = 5x2 − 5
k = a2
5x2 − 16x + 3 = 0
5x – 3
x
–
k = 52
∴ k = 25
1
(c) 2(x + y) = 22
(5x − 1)(x − 3) = 0
x + y = 11
5x − 1 = 0
y = 11 − x
5x = 1 or x − 3 = 0
1
x = __ or
x=3
5
Q.7.
y
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
–1 –0.5–1 0 0.5
–2
–3
(a)
x(11 − x) = 30
11x − x2 = 30
−x2 + 11x − 30 = 0
x2 − 11x + 30 = 0
(x − 5)(x − 6) = 0
∴x=5
y=6
1
1.5
2
2.5
3
3.5
4
4.5
or y = 5
Width = 6 m
5
Q. 8. (a)
Q
or x = 3
or x = 3.5
x(2x − 7) = 0
or x =
(i) x = ±3
(ii) x = 0
7
__
2
B with P
C with R
D with Q
22
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
or x = 6
(iii) x ≈ −2.5
(b) A with S
(iv) 2x2 − 7x = 0
x=0
or x = 6
∴ Length = 5 m
x
(i) x = 0.5
(iii) x = 0
xy = 30
or x = 4
(c) (i)
2x − 7 = y
3x +
x(2x − 7) = 15
(2x + 3)(x − 5) = 0
3
x = −__ or x = 5
2
y = −10 or y = 3
(
3
__
∴ − ,−10
2
(ii) x = 2y − 1
)
or
–
x
2x2 − 7x − 15 = 0
2
3
(3x + 2)(x − 3) = 0
3x + 2 = 0 or x − 3 = 0
3x = −2 or
2
x = −__
3
∴x=3
400
400
(c) ____ − _____ = 5
x+4
x
(5,3)
400(x + 4) − 400x = 5x(x + 4)
400x + 1,600 − 400x = 5x2 + 20x
(2y − 1)2 − (2y − 1)(y) = 10
4y2 − 4y + 1 − 2y2 + y − 10 = 0
5x2 + 20x − 1,600 = 0
2y2 − 3y − 9 = 0
x2 + 4x − 320 = 0
x + 20
(2y + 3)(y − 3) = 0
y=−
3
__
or y = 3
2
x = −4 or x = 5
(
3
∴ −4,− __
2
(iii) x = 3 − 2y
)
or
x=3
x –
16
x + 20 = 0
or x − 16 = 0
x = −20
(5,3)
or
x = 16
∴ x = 16
(3 − 2y)2 + y2 = 2
9 − 12y + 4y2 + y2 − 2 = 0
Q. 10. (a)
Q
(i)
y
5y2 − 12y + 7 = 0
(5y − 7)(y − 1) = 0
7
y = __ or y = 1
5
1
x = __ or x = 1
5
1 __
7
__
∴ ,
or (1,1)
55
x
( )
(ii)
y
1
Q. 9. (a) x + __
x = 2.9
x2 + 1 = 2.9x
x2 − 2.9x + 1 = 0
(× 10)
x
10x2 − 29x + 10 = 0
5x – 2 (5x − 2)(2x − 5) = 0
2x –
5
5x − 2 = 0
5x = 2
or
2x − 5 = 0
2
x = __ or
5
2x = 5
5
x = __
2
(iii)
y
30
20
(b) ___ + _____ = 15
x
x+1
30(x + 1) + 20(x) = 15x(x + 1)
30x + 30 + 20x = 15x2 + 15x
15x2 − 35x − 30 = 0
3x2 − 7x − 6 = 0
x
(b) (i) (−3,−4) and (3,4)
(ii) (−5,−1) and (5,1)
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
23
(b) Using algebra:
4x
(i) y = ___
3
16x2
2
x + ____ = 25
9
9x2 + 16x2 = 225
(ii) At a point of intersection,
for the specific x-value at that
point, each function, f and g,
takes the same value (i.e. has
the same y-value).
(iii) x2 + 8x + 12 = 2x2 + 14x + 15
25x2 = 225
0 = x2 + 6x + 3
___
x2 = 9
–6 ± √ 24
∴ x = _________ = –0.55
2
or –5.45
x = ±3
If x = 3, y = 4
If x = −3, y = −4
Test of x = –0.55
∴ (3,4)
f(–0.55) ≈ 7.9 g(–0.55) ≈ 7.9
or
(−3,−4)
(ii) x = 5y
25y2
−
5y2
y2
= 21
21y2
= 21
+
∴ Same y value
__ for each function
at x = –3 + √ 6. Similarly, the
functions f and g take
__ the same
value of x = –3 –√ 6.
y2 = 1
Q. 11. (a) 80 = 40t − 5t2
Q
y = ±1
5t2 − 40t + 80 = 0
y = 1, x = 5
t2 − 8t + 16 = 0
y = −1, x = −5
∴ (−5,−1)
(c)
or
(t − 4)2 = 0
(5,1)
t = 4 seconds
y
g
(b)
14
x + y = 27 ⇒ y = 27 − x
12
x2 − y2 = 9
10
(–0.55,7.9) 8
f
x2 − (27 − x)2 = 9
6
4
2
0
–7 –6 –5 –4 –3 –2 –1
0 1
–2
(–5.45,–1.9)
–4
x2 − (729 − 54x + x2) = 9
54x − 729 = 9
x
2
54x = 738
2
x = 13 __
3
1
y = 13 __
3
–6
–8
Q. 12. (a) Complete
l
the
h table
bl b
below:
l
Time in seconds, t
0
20
40
60
80
100
Altitude in km, h
45
57
65
69
69
65
7
1
h = 45 + ___ t – ____ t2
200
10
Substitute t = 0, 20, 40, 60, 80 and 100 into the above formula.
This gives h = 45, 57, 65, 69, 69 and 65.
24
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
(b) Graph of attitude over time.
y
70
60
50
40
30
20
10
0
x
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
(c) Use your graph to estimate the greatest altitude reached by the rockets.
Answer: Approximately 69.5 km
(d) Answer: After 26 seconds
(e)
7
1
h = 45 + ___ t – ____t2
200
10
1
7
∴ h = 45 + ___ (26) – ____ (26)2
200
10
h = 59.82 km
1
7
(f ) 45 + ___ t – ____ t2 = 9
200
10
9000 + 140t – t2 = 1800
t2 – 140t – 7200 = 0
(t – 180)(t + 40) = 0
t = 180 secs
t ≠ –40
(g) t = 180 seconds gives an altitude of 9 km
1
7
t = 181 ⇒ h = 45 + ___ (181) – ____ (181)2 ⇒ h = 7.895 km
200
10
∴ 9 – 7.895 = 1.105 km ⇒ Speed = 1.105 km/sec ≈ 1.1 km/sec
h’ =
7
___
10
–
t
____
100
OR
⇒ h’(180) = –1.1 ⇒ Speed = 1.1 km/sec
Active Maths 3 Book 1 – Strands 1–5 – Ch 7 Solutions
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