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

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 Page 1


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
Page 2


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
Page 3


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
 
 
Discriminant D is given by:
2
2 2
2 2
4
4 1
4 0
D b a c
p q
p q
0 D for all real values of p and q.
Thus, the roots of the equation are real.
4. For what values of k are the roots of the quadratic equation 
2
3 2 27 0 x k x real and 
equal?
Sol:
Given:
2
3 2 27 0 x k x
Here,
3, 2 27 a b k a n d c
It is given that the roots of the equation are real and equal; therefore, we have:
2
2
0
2 4 3 27 0
4 324 0
D
k
k
2
2
4 324
81
9
9 9
k
k
k
k o r k
5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and 
equal.
Sol:
The given equation is
2
2 5 10 0
2 5 10 0
k x x
k x k x
This is of the form 
2
0, a x b x c where , 2 5 10. a k b k a n d c
2
2 2
4 2 5 4 10 20 40 D b a c k k k k
The given equation will have real and equal roots if 0. D
Page 4


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
 
 
Discriminant D is given by:
2
2 2
2 2
4
4 1
4 0
D b a c
p q
p q
0 D for all real values of p and q.
Thus, the roots of the equation are real.
4. For what values of k are the roots of the quadratic equation 
2
3 2 27 0 x k x real and 
equal?
Sol:
Given:
2
3 2 27 0 x k x
Here,
3, 2 27 a b k a n d c
It is given that the roots of the equation are real and equal; therefore, we have:
2
2
0
2 4 3 27 0
4 324 0
D
k
k
2
2
4 324
81
9
9 9
k
k
k
k o r k
5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and 
equal.
Sol:
The given equation is
2
2 5 10 0
2 5 10 0
k x x
k x k x
This is of the form 
2
0, a x b x c where , 2 5 10. a k b k a n d c
2
2 2
4 2 5 4 10 20 40 D b a c k k k k
The given equation will have real and equal roots if 0. D
 
 
2
20 40 0
20 2 0
0 2 0
0 2
k k
k k
k o r k
k o r k
But, for 0, k we get 10 0, which is not true
Hence, 2 is the required value of k.
6. For what values of p are the  roots of the equation 
2
4 3 0. x p x real and equal?
Sol:
The given equation is 
2
4 3 0. x p x
This is of the form 
2
0, a x b x c where 4, 3. a b p a n d c
2 2 2
4 4 4 3 48 D b a c p p
The given equation will have real and equal roots if 0. D
2
2
48 0
48
48 4 3
p
p
p
Hence, 4 3 and 4 3 are the required values of p.
7. Find the nonzero value of k for which the roots of the quadratic equation 
2
9 3 0. x k x k
are real and equal.
Sol:
The given equation is 
2
9 3 0. x k x k
This is of the form 
2
0, a x b x c where 9, 3 . a b k a n d c k
2
2 2
4 3 4 9 9 36 D b a c k k k k
The given equation will have real and equal roots if 0. D
2
9 36 0
9 4 0
0 4 0
0 4
k k
k k
k o r k
k o r k
But, 0 k (Given)
Hence, the required values of k is 4.
8. Find the values of k for which the quadratic equation 
2
3 1 2 1 1 0. k x k x has real 
and equal roots.
Sol:
Page 5


1. Find the nature of roots of the following quadratic equations:
(i) 
2
2 8 5 0. x x
(ii) 
2
3 2 6 2 0. x x
(iii) 
2
5 4 1 0. x x
(iv) 5 2 6 0 x x
(v)
2
12 4 15 5 0 x x
(vi) 
2
2 0. x x
Sol:
(i) The given equation is
2
2 8 5 0. x x
This is of the form
2
0, a x b x c where 2, 8 5. a b a n d c
Discriminant, 
2
2
4 8 4 2 5 64 40 24 0 D b a c
Hence, the given equation has real and unequal roots.
(ii) The given equation is
2
3 2 6 2 0. x x
This is of the form
2
0, a x b x c where 3, 2 6 2. a b a n d c
Discriminant, 
2
2
4 2 6 4 3 2 24 24 0 D b a c
Hence, the given equation has real and equal roots.
(iii) The given equation is
2
5 4 1 0. x x
This is of the form
2
0, a x b x c where 5, 4 1. a b a n d c
Discriminant, 
2
2
4 4 4 5 1 16 20 4 0 D b a c
Hence, the given equation has no real roots.
(iv) The given equation is
 
 
2
5 2 6 0
5 10 6 0
x x
x x
This is of the form 
2
0, a x b x c where 5, 10 6. a b a n d c
Discriminant, 
2
2
4 10 4 5 6 100 120 20 0 D b a c
Hence, the given equation has no real roots.
(v) The given equation is 
2
12 4 15 5 0 x x
This is of the form 
2
0, a x b x c where 12, 4 15 5. a b a n d c
Discriminant, 
2
2
4 4 15 4 12 5 240 240 0 D b a c
Hence, the given equation has real and equal roots.
(vi) The given equation is 
2
2 0. x x
This is of the form 
2
0, a x b x c where 1, 1 2. a b a n d c
Discriminant, 
2
2
4 1 4 1 2 1 8 7 0 D b a c
Hence, the given equation has no real roots.
2. If a and b are distinct real numbers, show that the quadratic equations
2 2 2
2 2 1 0. a b x a b x has no real roots.
Sol:
The given equation is 
2 2 2
2 2 1 0. a b x a b x
2
2 2
2 2 2 2
2 2 2 2
2 2
2 2
2
2 4 2 1
4 2 8
4 8 4 8 8
4 8 4
4 2
4 0
D a b a b
a a b b a b
a a b b a b
a a b b
a a b b
a b
Hence, the given equation has no real roots.
3. Show that the roots of the equation 
2 2
0 x p x q are real for all real values of p and q.
Sol:
Given:
2 2
0 x p x q
Here,
2
1, a b p a n d c q
 
 
Discriminant D is given by:
2
2 2
2 2
4
4 1
4 0
D b a c
p q
p q
0 D for all real values of p and q.
Thus, the roots of the equation are real.
4. For what values of k are the roots of the quadratic equation 
2
3 2 27 0 x k x real and 
equal?
Sol:
Given:
2
3 2 27 0 x k x
Here,
3, 2 27 a b k a n d c
It is given that the roots of the equation are real and equal; therefore, we have:
2
2
0
2 4 3 27 0
4 324 0
D
k
k
2
2
4 324
81
9
9 9
k
k
k
k o r k
5. For what value of k are the roots of the quadratic equation 2 5 10 0 k x x real and 
equal.
Sol:
The given equation is
2
2 5 10 0
2 5 10 0
k x x
k x k x
This is of the form 
2
0, a x b x c where , 2 5 10. a k b k a n d c
2
2 2
4 2 5 4 10 20 40 D b a c k k k k
The given equation will have real and equal roots if 0. D
 
 
2
20 40 0
20 2 0
0 2 0
0 2
k k
k k
k o r k
k o r k
But, for 0, k we get 10 0, which is not true
Hence, 2 is the required value of k.
6. For what values of p are the  roots of the equation 
2
4 3 0. x p x real and equal?
Sol:
The given equation is 
2
4 3 0. x p x
This is of the form 
2
0, a x b x c where 4, 3. a b p a n d c
2 2 2
4 4 4 3 48 D b a c p p
The given equation will have real and equal roots if 0. D
2
2
48 0
48
48 4 3
p
p
p
Hence, 4 3 and 4 3 are the required values of p.
7. Find the nonzero value of k for which the roots of the quadratic equation 
2
9 3 0. x k x k
are real and equal.
Sol:
The given equation is 
2
9 3 0. x k x k
This is of the form 
2
0, a x b x c where 9, 3 . a b k a n d c k
2
2 2
4 3 4 9 9 36 D b a c k k k k
The given equation will have real and equal roots if 0. D
2
9 36 0
9 4 0
0 4 0
0 4
k k
k k
k o r k
k o r k
But, 0 k (Given)
Hence, the required values of k is 4.
8. Find the values of k for which the quadratic equation 
2
3 1 2 1 1 0. k x k x has real 
and equal roots.
Sol:
 
 
The given equation is 
2
3 1 2 1 1 0. k x k x
This is of the form 
2
0, a x b x c where 3 1, 2 1 1. a k b k a n d c
2
2
2
2
2
4
2 1 4 3 1 1
4 2 1 4 3 1
4 8 4 12 4
4 4
D b ac
k k
k k k
k k k
k k
The given equation will have real and equal roots if 0. D
2
4 4 0
4 1 0
0 1 0
0 1
k k
k k
k o r k
k o r k
Hence, 0 and 1are the required values of k.
9. Find the value of p for which the quadratic equation 
2
2 1 7 2 7 3 0. p x p x p
has real and equal roots.
Sol:
The given equation is 
2
2 1 7 2 7 3 0. p x p x p
This is of the form 
2
0, a x b x c where 2 1, 7 2 7 3. a p b p a n d c p
2
4 D b a c
2
7 2 4 2 1 7 3 p p p
2 2
49 28 4 4 14 3 p p p p
2 2
49 28 4 56 4 12 p p p p
2
7 24 16 p p
The given equation will have real and equal roots if 0. D
2
7 24 16 0 p p
2
7 24 16 0 p p
2
7 28 4 16 0 p p p
7 4 4 4 0 p p p
4 7 4 0 p p
4 0 7 4 0 p o r p
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118 videos|463 docs|105 tests

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

1. What are Quadratic Equations?
Ans. Quadratic equations are polynomial equations of the second degree, which means the highest power of the variable is 2. They are written in the form ax^2 + bx + c = 0, where a, b, and c are constants and 'a' cannot be zero.
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 method to be used depends on the complexity of the equation and the available information.
3. What is the quadratic formula?
Ans. The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0. It states that the solutions for x can be found using the formula: x = (-b ± √(b^2 - 4ac)) / 2a.
4. Can quadratic equations have more than two solutions?
Ans. No, quadratic equations can have at most two solutions. This is because a quadratic equation of the form ax^2 + bx + c = 0 can be factored into (x - x1)(x - x2) = 0, where x1 and x2 are the solutions. Therefore, there can be at most two values of x that satisfy the equation.
5. What is the discriminant of a quadratic equation?
Ans. The discriminant of a quadratic equation is the part of the quadratic formula that is inside the square root, i.e., b^2 - 4ac. It helps determine the nature of the solutions. If the discriminant is greater than zero, the equation has two distinct real solutions. If the discriminant is equal to zero, the equation has one real solution. If the discriminant is less than zero, the equation has two complex conjugate solutions.
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