RS Aggarwal Solutions: Exercise 4B - Quadratic Equations Class 10 Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : RS Aggarwal Solutions: Exercise 4B - Quadratic Equations Class 10 Notes | EduRev

 Page 1


1.
2
6 x x 3 0
Sol: 
2
2
6 3 0
6
x x
x x
2
3
2 2
2 x x 3 3 3 3 (Adding 
2
3 on both sides)
2
x 3 3 9 6
3 6 x (Taking square root on the both sides) 
Page 2


1.
2
6 x x 3 0
Sol: 
2
2
6 3 0
6
x x
x x
2
3
2 2
2 x x 3 3 3 3 (Adding 
2
3 on both sides)
2
x 3 3 9 6
3 6 x (Taking square root on the both sides) 
 
3 6 3 6
3 6 3 6
x o r x
x o r x
Hence, 3 6 and 3 6 are the roots of the given equation.
2.
2
4 1 0 x x
Sol:
2
2
4 1 0
4 1
x x
x x
2 2 2
2 2 2 1 2 x x (Adding 
2
2 on both sides)
2
2 1 4 3 x
2 3 x (Taking square root on the both sides)
2 3 2 3
2 3 2 3
x o r x
x o r x
Hence, 2 3 and 2 3 are the roots of the given equation.
3.
2
8 2 0 x x
Sol:
2
2
8 2 0
8 2
x x
x x
2 2 2
2 4 4 2 4 x x (Adding 
2
4 on both sides)
2
4 2 16 18 x
4 18 3 2 x (Taking square root on the both sides)
4 3 2 4 3 2
4 3 2 4 3 2
x or x
x or x
Hence, 4 3 2 and 4 3 2 are the roots of the given equation.
4.
2
4 4 3 3 0 x x
Sol:
2
2
4 4 3 3 0
4 4 3 3
x x
x x
2 2
2
2 2 2 3 3 3 3 x x [Adding 
2
3 on both sides]
Page 3


1.
2
6 x x 3 0
Sol: 
2
2
6 3 0
6
x x
x x
2
3
2 2
2 x x 3 3 3 3 (Adding 
2
3 on both sides)
2
x 3 3 9 6
3 6 x (Taking square root on the both sides) 
 
3 6 3 6
3 6 3 6
x o r x
x o r x
Hence, 3 6 and 3 6 are the roots of the given equation.
2.
2
4 1 0 x x
Sol:
2
2
4 1 0
4 1
x x
x x
2 2 2
2 2 2 1 2 x x (Adding 
2
2 on both sides)
2
2 1 4 3 x
2 3 x (Taking square root on the both sides)
2 3 2 3
2 3 2 3
x o r x
x o r x
Hence, 2 3 and 2 3 are the roots of the given equation.
3.
2
8 2 0 x x
Sol:
2
2
8 2 0
8 2
x x
x x
2 2 2
2 4 4 2 4 x x (Adding 
2
4 on both sides)
2
4 2 16 18 x
4 18 3 2 x (Taking square root on the both sides)
4 3 2 4 3 2
4 3 2 4 3 2
x or x
x or x
Hence, 4 3 2 and 4 3 2 are the roots of the given equation.
4.
2
4 4 3 3 0 x x
Sol:
2
2
4 4 3 3 0
4 4 3 3
x x
x x
2 2
2
2 2 2 3 3 3 3 x x [Adding 
2
3 on both sides]
 
 
2
2 3 3 3 0
2 3 0
3
2
x
x
x
Hence, 
3
2
is the repeated root of the given equation.
5.
2
2 5 3 0 x x
Sol:
2
2 5 3 0 x x
2
4 10 6 0 x x (Multiplying both sides by 2)
2
4 10 6 x x
2 2
2 5 5 5
2 2 2 6
2 2 2
x x [Adding 
2
5
2
on both sides]
2 2
5 25 24 25 49 7
2 6
2 4 4 4 2
x
5 7
2
2 2
x (Taking square root on both sides)
5 7 5 7
2 2
2 2 2 2
x o r x
7 5 2
2 1
2 2 3
x or
7 5 12
2 6
2 2 2
x
1
2
x or 3 x
Hence, 
1
2
and 3 are the roots of the given equation. 
6.
2
3 2 0 x x
Sol:
2
3 2 0 x x
2
9 3 6 0 x x (Multiplying both sides by 3)
2
9 3 6 x x
2 2
2 1 1 1
3 2 3 6
2 2 2
x x [Adding 
2
1
2
on both sides]
Page 4


1.
2
6 x x 3 0
Sol: 
2
2
6 3 0
6
x x
x x
2
3
2 2
2 x x 3 3 3 3 (Adding 
2
3 on both sides)
2
x 3 3 9 6
3 6 x (Taking square root on the both sides) 
 
3 6 3 6
3 6 3 6
x o r x
x o r x
Hence, 3 6 and 3 6 are the roots of the given equation.
2.
2
4 1 0 x x
Sol:
2
2
4 1 0
4 1
x x
x x
2 2 2
2 2 2 1 2 x x (Adding 
2
2 on both sides)
2
2 1 4 3 x
2 3 x (Taking square root on the both sides)
2 3 2 3
2 3 2 3
x o r x
x o r x
Hence, 2 3 and 2 3 are the roots of the given equation.
3.
2
8 2 0 x x
Sol:
2
2
8 2 0
8 2
x x
x x
2 2 2
2 4 4 2 4 x x (Adding 
2
4 on both sides)
2
4 2 16 18 x
4 18 3 2 x (Taking square root on the both sides)
4 3 2 4 3 2
4 3 2 4 3 2
x or x
x or x
Hence, 4 3 2 and 4 3 2 are the roots of the given equation.
4.
2
4 4 3 3 0 x x
Sol:
2
2
4 4 3 3 0
4 4 3 3
x x
x x
2 2
2
2 2 2 3 3 3 3 x x [Adding 
2
3 on both sides]
 
 
2
2 3 3 3 0
2 3 0
3
2
x
x
x
Hence, 
3
2
is the repeated root of the given equation.
5.
2
2 5 3 0 x x
Sol:
2
2 5 3 0 x x
2
4 10 6 0 x x (Multiplying both sides by 2)
2
4 10 6 x x
2 2
2 5 5 5
2 2 2 6
2 2 2
x x [Adding 
2
5
2
on both sides]
2 2
5 25 24 25 49 7
2 6
2 4 4 4 2
x
5 7
2
2 2
x (Taking square root on both sides)
5 7 5 7
2 2
2 2 2 2
x o r x
7 5 2
2 1
2 2 3
x or
7 5 12
2 6
2 2 2
x
1
2
x or 3 x
Hence, 
1
2
and 3 are the roots of the given equation. 
6.
2
3 2 0 x x
Sol:
2
3 2 0 x x
2
9 3 6 0 x x (Multiplying both sides by 3)
2
9 3 6 x x
2 2
2 1 1 1
3 2 3 6
2 2 2
x x [Adding 
2
1
2
on both sides]
 
 
2 2
1 1 25 5
3 6
2 4 4 2
x
1 5
3
2 2
x (Taking square root on both sides)
1 5 1 5
3 3
2 2 2 2
x or x
5 1 6
3 3
2 2 2
x or
5 1 4
3 2
2 2 2
x
2
1
3
x o r x
Hence, 1 and 
2
3
are the roots of the given equation.
7.
2
8 14 15 0 x x
Sol:
2
8 14 15 0 x x
2
16 28 30 0 x x (Multiplying both sides by 2)
2
16 28 30 x x
2 2
2 7 7 7
4 2 4 30
2 2 2
x x [Adding 
2
7
2
on both sides]
2 2
7 49 169 13
4 30
2 4 4 2
x
7 13
4
2 2
x (Taking square root on both sides)
7 13 7 13
4 4
2 2 2 2
x o r x
13 7 20
4 10
2 2 2
x or
13 7 6
4 3
2 2 2
x
5 3
2 4
x or x
Hence, 
5
2
and 
3
4
are the roots of the given equation.
8.
2
7 3 4 0 x x
Sol:
2
7 3 4 0 x x
2
49 21 28 0 x x (Multiplying both sides by 7)
Page 5


1.
2
6 x x 3 0
Sol: 
2
2
6 3 0
6
x x
x x
2
3
2 2
2 x x 3 3 3 3 (Adding 
2
3 on both sides)
2
x 3 3 9 6
3 6 x (Taking square root on the both sides) 
 
3 6 3 6
3 6 3 6
x o r x
x o r x
Hence, 3 6 and 3 6 are the roots of the given equation.
2.
2
4 1 0 x x
Sol:
2
2
4 1 0
4 1
x x
x x
2 2 2
2 2 2 1 2 x x (Adding 
2
2 on both sides)
2
2 1 4 3 x
2 3 x (Taking square root on the both sides)
2 3 2 3
2 3 2 3
x o r x
x o r x
Hence, 2 3 and 2 3 are the roots of the given equation.
3.
2
8 2 0 x x
Sol:
2
2
8 2 0
8 2
x x
x x
2 2 2
2 4 4 2 4 x x (Adding 
2
4 on both sides)
2
4 2 16 18 x
4 18 3 2 x (Taking square root on the both sides)
4 3 2 4 3 2
4 3 2 4 3 2
x or x
x or x
Hence, 4 3 2 and 4 3 2 are the roots of the given equation.
4.
2
4 4 3 3 0 x x
Sol:
2
2
4 4 3 3 0
4 4 3 3
x x
x x
2 2
2
2 2 2 3 3 3 3 x x [Adding 
2
3 on both sides]
 
 
2
2 3 3 3 0
2 3 0
3
2
x
x
x
Hence, 
3
2
is the repeated root of the given equation.
5.
2
2 5 3 0 x x
Sol:
2
2 5 3 0 x x
2
4 10 6 0 x x (Multiplying both sides by 2)
2
4 10 6 x x
2 2
2 5 5 5
2 2 2 6
2 2 2
x x [Adding 
2
5
2
on both sides]
2 2
5 25 24 25 49 7
2 6
2 4 4 4 2
x
5 7
2
2 2
x (Taking square root on both sides)
5 7 5 7
2 2
2 2 2 2
x o r x
7 5 2
2 1
2 2 3
x or
7 5 12
2 6
2 2 2
x
1
2
x or 3 x
Hence, 
1
2
and 3 are the roots of the given equation. 
6.
2
3 2 0 x x
Sol:
2
3 2 0 x x
2
9 3 6 0 x x (Multiplying both sides by 3)
2
9 3 6 x x
2 2
2 1 1 1
3 2 3 6
2 2 2
x x [Adding 
2
1
2
on both sides]
 
 
2 2
1 1 25 5
3 6
2 4 4 2
x
1 5
3
2 2
x (Taking square root on both sides)
1 5 1 5
3 3
2 2 2 2
x or x
5 1 6
3 3
2 2 2
x or
5 1 4
3 2
2 2 2
x
2
1
3
x o r x
Hence, 1 and 
2
3
are the roots of the given equation.
7.
2
8 14 15 0 x x
Sol:
2
8 14 15 0 x x
2
16 28 30 0 x x (Multiplying both sides by 2)
2
16 28 30 x x
2 2
2 7 7 7
4 2 4 30
2 2 2
x x [Adding 
2
7
2
on both sides]
2 2
7 49 169 13
4 30
2 4 4 2
x
7 13
4
2 2
x (Taking square root on both sides)
7 13 7 13
4 4
2 2 2 2
x o r x
13 7 20
4 10
2 2 2
x or
13 7 6
4 3
2 2 2
x
5 3
2 4
x or x
Hence, 
5
2
and 
3
4
are the roots of the given equation.
8.
2
7 3 4 0 x x
Sol:
2
7 3 4 0 x x
2
49 21 28 0 x x (Multiplying both sides by 7)
 
 
2
49 21 28 x x
2 2
2 3 3 3
7 2 7 28
2 2 2
x x [Adding 
2
3
2
on both sides]
2 2
3 9 121 11
7 28
2 4 4 2
x
3 11
7
2 2
x (Taking square root on both sides)
3 11 3 11
7 7
2 2 2 2
x or x
11 3 8
7 4
2 2 2
x or
11 3 14
7 7
2 2 2
x
4
1
7
x o r x
Hence, 
4
7
and 1 are the roots of the given equation.
9.
2
3 2 1 0 x x
Sol:
2
3 2 1 0 x x
2
9 6 3 0 x x (Multiplying both sides by 3)
2
9 6 3 x x
2
2 2
3 2 3 1 1 3 1 x x [Adding 
2
1 on both sides]
2 2
3 1 3 1 4 2 x
3 1 2 x (Taking square root on both sides)
3 1 2 x or 3 1 2 x
3 3 x or 3 1 x
1 x or
1
3
x
Hence, 1 and 
1
3
are the roots of the given equation.
10.
2
5 6 2 0 x x
Sol:
2
5 6 2 0 x x
2
25 30 10 0 x x (Multiplying both sides by 5)
2
25 30 10 x x
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