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Form 2 Mathematics Quadratic Expressions and Equations Questions and Answers
Solve each of the following equations:
(a)
x
2
-
25
=
0
(b)
3
x
2
-
2
x
=
5
(c)
3
x
3
+
x
2
–
4
x
=
0
(d)
1
-
6
x
2
=
x
(6m 4s)
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1.
The square of a number is 4 more than three times the number. Find the number.
2.
Given that the lengths of the three sides a right-angled triangle are x, x+ 1 and x + 2 units. Find the value of x.
3.
A triangle ABC has a base of (x + 3) cm and a height of x cm. If its area is 5
c
m
2
. Calculate the length of its base.
4.
After reducing by an equal amount the length and width of a rectangle which were originally 8 cm and 5 cm respectively, the area of the new rectangle is 18
c
m
2
. Find the dimensions of the new rectangle.
5.
A number of people bought a 300-hectare farm which they shared equally. If the number of hectares per person was 5 less than their number, find the number of people.
6.
The figure below shows two concentric circles. If the ratio of their radii is 1:2 and the area of the shaded region is 75
c
m
2
, calculate the area of the larger circle.
7.
A picture measuring 5 cm by 8 cm is mounted on a frame so as to leave a uniform margin of width x cm all round. If the area of the margin is 30
c
m
2
, find x.
8.
The figure below shows a rectangular lawn of dimensions 6x metres by x metres. In the centre is a rectangular flower garden of length (x + 4) m and width (x - 1) m. If the area of the shaded region is 40
m
2
, calculate the area of the flower garden.
9.
There are 240 exercise books to be given to a class. Thirty-four students in the class did not receive the books for various reasons. If the books were then shared equally among the remaining members of the class, the number of books given to each student is equal to the number of students in the class. Find the number of students in class.
10.
Members of a group decided to raise K£ 100 towards a charity by contributing equal amount. Five of them were unable to contribute. The rest had, therefore, to pay K£ 1 more each to realise the same amount. How many members were in the group originally?
11.
ABC is an isosceles triangle in which AB = AC. The size of angles ABC and ACB are (3x^2- 2x + 4)° and (9x - 6)° respectively. Calculate the two possible sets of the three angles of triangle ABC.
12.
Expand and simplify each of the following expressions: (a) (6x + 2) (4x + 3) (b) (3x - 2) (x + 5) (c) (x -1)(x + 2) (d) (2x - 3)(x + 2) (e) (6x - 2)(2x + 8) (f) (2 - x) (x + 4)
13.
Expand and simplify each of the following expressions. (a) (x +7) (x + 8) (b) (3x - 2) (2x + 3) (c) (5x + 3)(3x + 5) (d) (7x - 4)(4x - 7) (e) (m - n)(n - m) (f) (2x -3)(x + 3)
14.
Expand and simplify each of the following expressions : (a) (ax + b)(2ax - 3b) (b) (4 - 2x)(
1
2
- x) (c) (6x - 5)(5x + 6) (d) (y - 2)(y + 2) (e) (x + 2a) (x + 2a) (f) (3x - 4)(5x - 6)
15.
Expand and simplify each of the following expressions (a) (x + 2a)(x - 2a) (b) (x - 7)(x + 7) (c)
(
x
+
2
)
^2 (d)
(
4
x
+
2
)
^2
16.
Expand and simplify each of the following expressions (a)
(
1
2
+
x
)
2
(b)
(
a
x
+
d
)
2
(c)
(
1
4
+
1
x
)
2
(d)
(
x
-
3
)
2
(e)
(
4
x
-
3
4
)
2
(f)
(
1
-
1
x
)
2
17.
Use the quadratic identities to write down the expansions of each of the following expressions: (a)
(
x
+
5
)
2
(b)
(
x
-
5
)
2
(c)
(
4
x
+
3
)
2
(d)
(
2
x
+
3
)
2
18.
Use the quadratic identities to write down the expansions of each of the following expressions. (a)
(
3
-
x
)
2
(b)
(
1
2
+
x
)
2
(c)
(
1
x
-
1
)
2
(d)
(
1
4
+
3
4
b
)
2
19.
Use the quadratic identities to write down the expansions of each of the following expressions : (a)
(
1
2
a
-
1
3
b
)
2
(b)
(
1
-
1
a
)
(
1
+
1
a
)
(c)
(
1
x
+
1
y
)
(
1
x
-
1
y
)
(d)
(
-
2
x
-
3
y
)
(
-
2
x
+
3
y
)
20.
Use the quadratic identities to write down the expansions of each of the following expressions: (a)
(
1
3
y
+
1
4
x
)
2
(b)
(
-
1
3
x
-
1
)
2
(c)
(
1
5
a
+
4
5
b
)
2
(d)
(
3
7
x
+
1
2
y
)
2
21.
Factorise each of the following expressions: (a)
x
2
+
6
x
+
8
(b)
x
2
-
5
x
+
6
(c)
x
2
+
4
x
-
21
(d)
x
2
+
x
-
2
(e)
x
2
+
2
x
–
35
(f)
x
2
-
10
x
+
24
22.
Factorise each of the following expressions. (a)
x
2
-
4
x
-
32
(b)
x
2
+
3
x
-
54
(c)
x
2
+
2
x
+
1
(d)
x
2
+
4
x
+
4
(e)
x
2
-
2
x
+
1
(f)
x
2
-
14
x
+
49
23.
Factorise each of the following expressions : (a)
x
2
-
16
x
+
64
(b)
x
2
-
4
(c)
1
-
x
2
(d)
(
x
2
-
16
)
(e)
x
2
-
5
x
+
6
(f)
x
2
-
5
x
-
6
24.
Factorise each of the following expressions (a)
x
2
+
2
x
+
a
x
+
2
a
(b)
t
2
+
8
t
+
12
25.
Factorise each of the following expressions: (a)
2
x
2
+
3
x
–
2
(b)
3
x
2
–
2
x
–
8
(c)
4
x
2
+
7
x
+
3
(d)
5
x
2
–
21
x
+
4
26.
Factorise each of the following expressions: (a)
14
x
2
–
16
x
+
2
(b)
3
x
2
+
11
x
+
6
(c)
8
–
2
x
–
3
x
2
(d)
16
x
+
24
+
2
x
2
27.
Factorise each of the following expressions: (a)
35
x
2
+
43
x
+
12
(b)
–
21
x
2
+
58
x
–
21
(c)
150
x
2
–
25
x
–
21
(d)
17
x
2
+
82
x
–
15
28.
Factorise each of the following expressions: (a)
8
x
2
+
6
x
+
1
(b)
x
2
–
4
x
–
117
(c)
9
x
2
+
48
x
+
64
(d)
75
x
2
+
10
x
–
21
29.
Factorise each of the following expressions: (a)
1
+
7
x
–
30
x
2
(b)
19
x
2
–
22
x
+
3
(c)
63
x
2
+
130
x
+
63
(
d
)
x
2
–
2
15
x
–
1
15
30.
Factorise each of the following expressions: (a)
4
x
2
–
12
x
+
9
(b)
9
x
2
+
6
x
+
1
(c)
2
x
2
–
32
y
2
(
d
)
1
–
2
y
+
1
y
2
31.
Factorise each of the following expressions: (a)
t
8
–
t
4
(b)
35
x
2
–
x
–
12
(c)
4
x
2
+
2
x
+
1
4
(d)
2
x
2
+
4
x
–
6
32.
Factorise each of the following expressions: (a)
21
x
2
+
17
x
–
4
(b)
25
x
2
–
5
2
x
+
1
16
(c)
26
x
–
29
x
2
+
3
(d)
1
16
x
2
–
7
2
x
+
49
33.
Factorise each of the following expressions: (a)
24
x
2
–
43
x
–
56
(b)
33
x
2
–
16
x
–
4
(c)
1
x
2
–
4
x
+
4
(d)
w
4
–
t
4
+
t
2
+
w
2
34.
Solve each of the following equations: (a)
x
2
+
6
x
+
8
=
0
(b)
2
x
2
+
3
x
-
2
=
0
(c)
14
x
2
-
16
x
+
2
=
0
(d)
4
x
2
+
7
x
-
5
=
x
2
-
9
35.
Solve each of the following equations: (a)
x
2
-
25
=
0
(b)
3
x
2
-
2
x
=
5
(c)
3
x
3
+
x
2
–
4
x
=
0
(d)
1
-
6
x
2
=
x
36.
Solve each of the following equations: (a)
4
x
2
-
x
-
1
2
=
0
(b)
x
+
20
x
=
9
(c)
16
x
+
24
+
2
x
2
=
0
(d)
1
4
-
1
9
x
2
=
0
37.
Solve each of the following equations: (a)
3
x
2
-
8
x
=
16
(b)
(
2
x
-
1
)
2
-
1
=
3
(c)
x
2
+
5
6
x
=
1
6
(
d
)
(
1
-
1
x
)
2
-
1
=
35
38.
Solve each of the following equations: (a)
1
3
x
2
=
2
x
-
3
(b)
-
12
x
=
-
x
+
4
(
c
)
x
2
y
2
-
x
2
-
y
4
+
y
2
=
0
39.
Solve each of the following equations: (a)
7
x
2
-
15
x
+
8
=
0
(b)
8
x
2
+
6
x
+
1
=
0
(c)
x
2
-
2
15
x
-
1
15
=
0
(d)
2
x
2
-
15
x
=
7
40.
Solve each of the following equations: (a)
x
2
2
+
1
=
9
x
4
(b)
1
-
x
=
1
x
+
3
(c)
x
2
-
1
18
=
x
6
(d)
4
(
1
y
2
)
-
4
(
1
y
)
+
1
=
0
41.
Solve each of the following equations: (a)
6
x
-
4
=
x
+
4
x
(b)
6
x
+
8
2
+
1
x
=
0
42.
For each of the following pairs of roots, obtain a corresponding equation in the form
a
x
2
+ bx +c = 0, where a, b and c are constants. (a) (-2, 2) (b)
(
1
2
,
1
3
,
)
(c)
(
1
4
,
-
3
)
(d) (-3, -5) (e) (0.5, 0.75)
43.
For each of the following pairs of roots, obtain a corresponding equation in the form
a
x
2
+ bx +c = 0, where a, b and c are constants. (a) (0, -3) (b) (-3, -3) (c)
(
-
2
1
2
,
31
2
)
(d) (p, q) (e)
(
1
p
,
1
q
)
×
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