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 Page 1


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
Page 2


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
Page 3


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
 
 
2
2 2
1 1
2 3
x x
x x
2
2 2 2
4 2 3 2
4 3 2
1 2 1 3
2 1 2 2 3
2 2 1 0
x x x x
x x x x x
x x x x
This is not of the form
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
2. Which of the following are the roots of 
2
3 2 1 0? x x
(i) 1 (ii)
1
3
(iii)
1
2
Sol:
The given equation is 
2
3 2 1 0 . x x
(i) 1 x
L.H.S.
2
2 1 x x
2
3 1 2 1 1
3 2 1
0
. . . R H S
Thus, 1 is a root of 
2
3 2 1 0 . x x
(ii) On subtracting 
1
3
x in the given equation, we get:
L.H.S. 
2
3 2 1 x x
2
1 1
3 2 1
3 3
1 2
3 1
9 3
1 2 3
3
0
3
0
. . . R H S
Thus,
1
3
is a root of
2
3 2 1 0 x x
Page 4


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
 
 
2
2 2
1 1
2 3
x x
x x
2
2 2 2
4 2 3 2
4 3 2
1 2 1 3
2 1 2 2 3
2 2 1 0
x x x x
x x x x x
x x x x
This is not of the form
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
2. Which of the following are the roots of 
2
3 2 1 0? x x
(i) 1 (ii)
1
3
(iii)
1
2
Sol:
The given equation is 
2
3 2 1 0 . x x
(i) 1 x
L.H.S.
2
2 1 x x
2
3 1 2 1 1
3 2 1
0
. . . R H S
Thus, 1 is a root of 
2
3 2 1 0 . x x
(ii) On subtracting 
1
3
x in the given equation, we get:
L.H.S. 
2
3 2 1 x x
2
1 1
3 2 1
3 3
1 2
3 1
9 3
1 2 3
3
0
3
0
. . . R H S
Thus,
1
3
is a root of
2
3 2 1 0 x x
 
 
(iii) On subtracting 
1
2
x in the given equation, we get
L.H.S.
2
3 2 1 x x
2
1 1
3 2 1
2 2
1
3 1 1
4
3
2
4
3 8
4
5
0
4
Thus, . . . . . L H S R H S
Hence, 
1
2
is a solution of 
2
3 2 1 0 . x x
3. Find the value of k for which 1 x is a root of the equation
2
3 0. x k x
Sol:
It is given that 1 x is a root of 
2
3 0 . x k x
Therefore, 1 x must satisfy the equation.
2
1 1 3 0
4 0
4
k
k
k
Hence, the required value of k is 4.
4. Find the value of a and b for which
3
4
x and 2 x are the roots of the equation
2
6 0 a x b x
Sol:
It is given that 
3
4
is a root of 
2
6 0; a x b x therefore, we have:
2
3 3
6 0
4 4
a b
Page 5


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
 
 
2
2 2
1 1
2 3
x x
x x
2
2 2 2
4 2 3 2
4 3 2
1 2 1 3
2 1 2 2 3
2 2 1 0
x x x x
x x x x x
x x x x
This is not of the form
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
2. Which of the following are the roots of 
2
3 2 1 0? x x
(i) 1 (ii)
1
3
(iii)
1
2
Sol:
The given equation is 
2
3 2 1 0 . x x
(i) 1 x
L.H.S.
2
2 1 x x
2
3 1 2 1 1
3 2 1
0
. . . R H S
Thus, 1 is a root of 
2
3 2 1 0 . x x
(ii) On subtracting 
1
3
x in the given equation, we get:
L.H.S. 
2
3 2 1 x x
2
1 1
3 2 1
3 3
1 2
3 1
9 3
1 2 3
3
0
3
0
. . . R H S
Thus,
1
3
is a root of
2
3 2 1 0 x x
 
 
(iii) On subtracting 
1
2
x in the given equation, we get
L.H.S.
2
3 2 1 x x
2
1 1
3 2 1
2 2
1
3 1 1
4
3
2
4
3 8
4
5
0
4
Thus, . . . . . L H S R H S
Hence, 
1
2
is a solution of 
2
3 2 1 0 . x x
3. Find the value of k for which 1 x is a root of the equation
2
3 0. x k x
Sol:
It is given that 1 x is a root of 
2
3 0 . x k x
Therefore, 1 x must satisfy the equation.
2
1 1 3 0
4 0
4
k
k
k
Hence, the required value of k is 4.
4. Find the value of a and b for which
3
4
x and 2 x are the roots of the equation
2
6 0 a x b x
Sol:
It is given that 
3
4
is a root of 
2
6 0; a x b x therefore, we have:
2
3 3
6 0
4 4
a b
 
 
9 3
6
16 4
9 12
6
16
9 12 96 0
a b
a b
a b
3 4 32 .......... a b i
Again, 2 is a root of 
2
6 0; a x b x therefore, we have:
2
2 2 6 0
4 2 6
2 3 .........
a b
a b
a b i i
On multiplying (ii) by 4 and adding the result with (i), we get:
3 4 8 4 32 12
11 44
a b a b
a
4 a
Putting the value of a in (ii), we get:
2 4 3
8 3
5
b
b
b
Hence, the required values of a and b are 4 and 5, respectively.
5. 2 3 3 1 0 x x
Sol:
2 3 3 1 0 x x
2 3 0 3 1 0
2 3 3 1
3 1
2 3
x or x
x or x
x or x
Hence the roots of the given equation are 
3 1
.
2 3
a n d
6.
2
4 5 0 x x
Sol:
2
4 5 0 x x
4 5 0 x x
0 4 5 0 x o r x

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FAQs on Quadratic Equations (Exercise 4A) RD Sharma Solutions - 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 represents an unknown variable. These equations have a maximum of two solutions, which can be found using various methods such as factoring, completing the square, or using the quadratic formula.

2. How do you solve quadratic equations by factoring?

Ans. To solve quadratic equations by factoring, follow these steps: 1. Write the quadratic equation in the form ax^2 + bx + c = 0. 2. Factor the equation into two binomials. 3. Set each binomial equal to zero and solve for x. 4. The solutions obtained are the values of x that satisfy the quadratic equation.

3. What is the quadratic formula and how is it used to solve quadratic equations?

Ans. The quadratic formula is given by x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. To use the quadratic formula, substitute the values of a, b, and c into the formula and simplify. The solutions obtained using the quadratic formula are the roots of the quadratic equation.

4. Can quadratic equations have more than two solutions?

Ans. No, quadratic equations can have a maximum of two solutions. This is because a quadratic equation is a second-degree polynomial, and a polynomial of degree n can have a maximum of n solutions. In the case of quadratic equations, the solutions can be real or complex, but there will always be at most two distinct solutions.

5. How can quadratic equations be used in real-life applications?

Ans. Quadratic equations have various real-life applications, including physics, engineering, finance, and computer graphics. For example, they can be used to model the trajectory of a projectile, calculate the maximum or minimum value of a quantity, analyze the profit or loss in a business, or design curved shapes in computer animations. By solving quadratic equations, we can find important information and make predictions in these practical scenarios.

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RD Sharma Solutions: Quadratic Equations (Exercise 4A) Class 10 Questions

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