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 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

	Mathematics (Maths) Class 10 127 videos\|551 docs\|75 tests

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FAQs on RS Aggarwal Solutions: Quadratic Equations (Exercise 4B) - Mathematics (Maths) Class 10

1. What are quadratic equations?

Ans. Quadratic equations are polynomial equations of degree 2, where the highest power of the variable is 2. They are represented in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

2. How do you solve quadratic equations?

Ans. Quadratic equations can be solved using various methods such as factoring, completing the square, or using the quadratic formula. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), where x represents the variable and a, b, and c are the coefficients in the equation ax^2 + bx + c = 0.

3. What is the discriminant of a quadratic equation?

Ans. The discriminant of a quadratic equation is the expression inside the square root in the quadratic formula, i.e., b^2 - 4ac. It helps determine the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (also known as a repeated root). And if the discriminant is negative, the equation has two complex roots.

4. Can quadratic equations have irrational or imaginary roots?

Ans. Yes, quadratic equations can have irrational or imaginary roots. If the discriminant (b^2 - 4ac) is negative, the roots of the quadratic equation will be complex or imaginary. If the discriminant is positive, the roots can be either rational or irrational, depending on the values of the coefficients.

5. Are there any real-life applications of quadratic equations?

Ans. Yes, quadratic equations have several real-life applications. They are used in various fields such as physics, engineering, finance, computer graphics, and architecture. Some examples include calculating projectile motion, designing parabolic satellite dishes, modeling the growth of populations, and optimizing profit in business scenarios.

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