RS Aggarwal Solutions: Quadratic Equations (Exercise 4C) | Mathematics (Maths) Class 10 PDF Download

 Page 1


1.
2
2 7 x x 6 0
Sol: 
(i)
2
Here, 
a 2,
7,
2 7 x x 6 0
b
c 6
Discriminant D is diven by:  
2
2
7 4 2 6
D b 4 a c
49 48
1
(ii)
2
3 2 x x 8 0
Here, 
Page 2


1.
2
2 7 x x 6 0
Sol: 
(i)
2
Here, 
a 2,
7,
2 7 x x 6 0
b
c 6
Discriminant D is diven by:  
2
2
7 4 2 6
D b 4 a c
49 48
1
(ii)
2
3 2 x x 8 0
Here, 
 
 
3,
2,
8
a
b
c
Discriminant D is given by:  
2
2
4
2 4 3 8
4 96
92
D b a c
(iii)
2
2 5 2 4 0 x x
Here,
2,
5 2,
4
a
b
c
Discriminant D is given by:
2
2
4
5 2 4 2 4
25 2 32
50 32
18
D b a c
(iv)
2
3 2 2 2 3 0 x x
Here,
3
2 2,
2 3
a
b
c
Discriminant D is given by:
2
2
4
2 2 4 3 2 3
4 2 8 3
8 24
32
D b a c
(v) 1 2 1 0 x x
2
2 3 1 0 x x
Comparing it with 
2
0, a x b x c we get
Page 3


1.
2
2 7 x x 6 0
Sol: 
(i)
2
Here, 
a 2,
7,
2 7 x x 6 0
b
c 6
Discriminant D is diven by:  
2
2
7 4 2 6
D b 4 a c
49 48
1
(ii)
2
3 2 x x 8 0
Here, 
 
 
3,
2,
8
a
b
c
Discriminant D is given by:  
2
2
4
2 4 3 8
4 96
92
D b a c
(iii)
2
2 5 2 4 0 x x
Here,
2,
5 2,
4
a
b
c
Discriminant D is given by:
2
2
4
5 2 4 2 4
25 2 32
50 32
18
D b a c
(iv)
2
3 2 2 2 3 0 x x
Here,
3
2 2,
2 3
a
b
c
Discriminant D is given by:
2
2
4
2 2 4 3 2 3
4 2 8 3
8 24
32
D b a c
(v) 1 2 1 0 x x
2
2 3 1 0 x x
Comparing it with 
2
0, a x b x c we get
 
 
2, 3 1 a b a n d c
Discriminant, 
2
2
4 3 4 2 1 9 8 1 D b a c
(vi)
2
1 2 x x
2
2 1 0 x x
Here,
2,
1,
1
a
b
c
Discriminant D is given by:
2
2
4
1 4 2 1
1 8
9
D b ac
Find the roots of the each of the following equations, if they exist, by applying the 
quadratic formula:
2.
2
4 1 0 x x
Sol:
Given:
2
4 1 0 x x
On comparing it with 
2
0, a x b x c we get:
1, 4 1 a b a n d c
Discriminant D is given by:
2
4 D b a c
2
4 4 1 1
16 4
20
20 0
Hence, the roots of the equation are real.
Roots and are given by:
2 2 5
4 20 4 2 5
2 5
2 2 1 2 2
2 2 5
4 20
4 2 5
2 5
2 2 2 2
b D
a
b D
a
Page 4


1.
2
2 7 x x 6 0
Sol: 
(i)
2
Here, 
a 2,
7,
2 7 x x 6 0
b
c 6
Discriminant D is diven by:  
2
2
7 4 2 6
D b 4 a c
49 48
1
(ii)
2
3 2 x x 8 0
Here, 
 
 
3,
2,
8
a
b
c
Discriminant D is given by:  
2
2
4
2 4 3 8
4 96
92
D b a c
(iii)
2
2 5 2 4 0 x x
Here,
2,
5 2,
4
a
b
c
Discriminant D is given by:
2
2
4
5 2 4 2 4
25 2 32
50 32
18
D b a c
(iv)
2
3 2 2 2 3 0 x x
Here,
3
2 2,
2 3
a
b
c
Discriminant D is given by:
2
2
4
2 2 4 3 2 3
4 2 8 3
8 24
32
D b a c
(v) 1 2 1 0 x x
2
2 3 1 0 x x
Comparing it with 
2
0, a x b x c we get
 
 
2, 3 1 a b a n d c
Discriminant, 
2
2
4 3 4 2 1 9 8 1 D b a c
(vi)
2
1 2 x x
2
2 1 0 x x
Here,
2,
1,
1
a
b
c
Discriminant D is given by:
2
2
4
1 4 2 1
1 8
9
D b ac
Find the roots of the each of the following equations, if they exist, by applying the 
quadratic formula:
2.
2
4 1 0 x x
Sol:
Given:
2
4 1 0 x x
On comparing it with 
2
0, a x b x c we get:
1, 4 1 a b a n d c
Discriminant D is given by:
2
4 D b a c
2
4 4 1 1
16 4
20
20 0
Hence, the roots of the equation are real.
Roots and are given by:
2 2 5
4 20 4 2 5
2 5
2 2 1 2 2
2 2 5
4 20
4 2 5
2 5
2 2 2 2
b D
a
b D
a
 
 
Thus, the roots of the equation are 2 5 and 2 5 .
3.
2
6 4 0 x x
Sol:
Given:
2
6 4 0 x x
On comparing it with 
2
0, a x b x c we get:
1, 6 4 a b a n d c
Discriminant D is given by:
2
4 D b a c
2
6 4 1 4
36 16
20 0
Hence, the roots of the equation are real.
Roots and are given by:
2 3 3
6 20 6 2 5
3 5
2 2 1 2 2
2 3 5
6 20
6 2 5
3 5
2 2 2 2
b D
a
b D
a
Thus, the roots of the equation are 3 2 5 and 3 2 5 .
4.
2
2 4 0. x x
Sol:
The given equation is 
2
2 4 0. x x
Comparing it with 
2
0, a x b x c we get
2, 1 a b and 4 c
Discriminant, 
2
2
4 1 4 2 4 1 32 33 0 D b a c
So, the given equation has real roots.
Now, 33 D
1 33 1 33
2 2 2 4
1 33 1 33
2 2 2 4
b D
a
b D
a
Page 5


1.
2
2 7 x x 6 0
Sol: 
(i)
2
Here, 
a 2,
7,
2 7 x x 6 0
b
c 6
Discriminant D is diven by:  
2
2
7 4 2 6
D b 4 a c
49 48
1
(ii)
2
3 2 x x 8 0
Here, 
 
 
3,
2,
8
a
b
c
Discriminant D is given by:  
2
2
4
2 4 3 8
4 96
92
D b a c
(iii)
2
2 5 2 4 0 x x
Here,
2,
5 2,
4
a
b
c
Discriminant D is given by:
2
2
4
5 2 4 2 4
25 2 32
50 32
18
D b a c
(iv)
2
3 2 2 2 3 0 x x
Here,
3
2 2,
2 3
a
b
c
Discriminant D is given by:
2
2
4
2 2 4 3 2 3
4 2 8 3
8 24
32
D b a c
(v) 1 2 1 0 x x
2
2 3 1 0 x x
Comparing it with 
2
0, a x b x c we get
 
 
2, 3 1 a b a n d c
Discriminant, 
2
2
4 3 4 2 1 9 8 1 D b a c
(vi)
2
1 2 x x
2
2 1 0 x x
Here,
2,
1,
1
a
b
c
Discriminant D is given by:
2
2
4
1 4 2 1
1 8
9
D b ac
Find the roots of the each of the following equations, if they exist, by applying the 
quadratic formula:
2.
2
4 1 0 x x
Sol:
Given:
2
4 1 0 x x
On comparing it with 
2
0, a x b x c we get:
1, 4 1 a b a n d c
Discriminant D is given by:
2
4 D b a c
2
4 4 1 1
16 4
20
20 0
Hence, the roots of the equation are real.
Roots and are given by:
2 2 5
4 20 4 2 5
2 5
2 2 1 2 2
2 2 5
4 20
4 2 5
2 5
2 2 2 2
b D
a
b D
a
 
 
Thus, the roots of the equation are 2 5 and 2 5 .
3.
2
6 4 0 x x
Sol:
Given:
2
6 4 0 x x
On comparing it with 
2
0, a x b x c we get:
1, 6 4 a b a n d c
Discriminant D is given by:
2
4 D b a c
2
6 4 1 4
36 16
20 0
Hence, the roots of the equation are real.
Roots and are given by:
2 3 3
6 20 6 2 5
3 5
2 2 1 2 2
2 3 5
6 20
6 2 5
3 5
2 2 2 2
b D
a
b D
a
Thus, the roots of the equation are 3 2 5 and 3 2 5 .
4.
2
2 4 0. x x
Sol:
The given equation is 
2
2 4 0. x x
Comparing it with 
2
0, a x b x c we get
2, 1 a b and 4 c
Discriminant, 
2
2
4 1 4 2 4 1 32 33 0 D b a c
So, the given equation has real roots.
Now, 33 D
1 33 1 33
2 2 2 4
1 33 1 33
2 2 2 4
b D
a
b D
a
 
 
Hence, 
1 33
4
and 
1 33
4
are the roots of the given equation.
5.
2
25 30 7 0 x x
Sol:
Given:
2
25 30 7 0 x x
On comparing it with 
2
0, a x b x x we get;
25, 30 7 a b a n d c
Discriminant D is given by:
2
2
4
30 4 25 7
D b a c
900 700
200
200 0
Hence, the roots of the equation are real.
Roots and are given by:
10 3 2 3 2
30 200 30 10 2
2 2 25 50 50 5
10 3 2 3 2
30 200 30 10 2
2 2 25 50 50 5
b D
a
b D
a
Thus, the roots of the equation are 
3 2 3 2
.
5 5
a n d
6.
2
16 24 1 x x
Sol:
Given:
2
2
16 24 1
16 24 1 0
x x
x x
On comparing it with 
2
0, a x b x x we get;
16, 24 1 a b a n d c
Discriminant D is given by:
2
4 D b a c
2
24 4 16 1