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RS Aggarwal Solutions: Quadratic Equations (Exercise 4C) - Mathematics (Maths) Class 10

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

1. What are quadratic equations?
Ans. Quadratic equations are algebraic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. These equations have a degree of 2 and can have two solutions, known as roots.
2. How do you solve quadratic equations?
Ans. Quadratic equations can be solved using various methods, such as factoring, completing the square, and using the quadratic formula. The most commonly used method is the quadratic formula, which states that the roots of a quadratic equation ax^2 + bx + c = 0 can be found using the formula x = (-b ± √(b^2 - 4ac)) / (2a).
3. What is the discriminant of a quadratic equation?
Ans. The discriminant of a quadratic equation is the expression b^2 - 4ac, which is found in the quadratic formula. It helps determine the nature of the roots of the equation. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root. And if it is negative, the equation has two complex roots.
4. Can quadratic equations have irrational roots?
Ans. Yes, quadratic equations can have irrational roots. When the discriminant of a quadratic equation is a perfect square, the roots can be expressed as irrational numbers. For example, the equation x^2 - 2 = 0 has the roots ±√2, which are irrational numbers.
5. How are quadratic equations used in real life?
Ans. Quadratic equations have various applications in real life. They are used in physics to describe the motion of objects under the influence of gravity, in engineering to model the behavior of mechanical systems, in finance to calculate compound interest, and in computer graphics to create smooth curves and animations.
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