Class 10 Maths · Chapter 4

Quadratic Equations

61 Class 10 Maths questions on Quadratic Equations from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 1 of 4

1. Q1 NCERT · Exercise 4.1
   Check whether the ​following ​are quadratic ​equations: ​

   1. (i)$(x+1{)}^2=2(x-3)$ ​
   2. (ii)$x^2-2x=(-2)(3-x)$
   3. (iii)$(x-2)(x+1)=(x-1)(x+3)$ ​
   4. (iv)$(x-3)(2x+1)=x(x+5)$ ​
   5. (v)$(2x-1)(x-3)=(x+5)(x-1)$
   6. (vi)$x^2+3x+1=(x-2{)}^2$
   7. (vii)$(x+2{)}^3=2x(x^2-1)$
   8. (viii)$x^3-4x^2-x+1=(x-2{)}^3$
   Show answer

   Quadratic: (i), (ii), (iv), (v), (viii). Not quadratic: (iii) And (vi) are linear, (vii) Is cubic.

   Step-by-step solution →
2. Q2 NCERT · Exercise 4.1
   Represent the following ​situation in the form ​of ​a quadratic equation: (i) The area of a rectangular ​plot is 528 m${}^2$. The ​length of the plot (in metres) is one more than ​twice its breadth. We need ​to find the length and breadth of the ​plot.
   Show answer

   $2x^2+x-528=0$, where $x$ is the breadth of the plot in metres.

   Step-by-step solution →
3. Q3 NCERT · Exercise 4.2
   Find the roots ​of ​the ​following quadratic equations ​by ​factorisation: ​

   1. (i)$x^2-3x-10=0$ ​
   2. (ii)$2x^2+x-6=0$ ​
   3. (iii)$\sqrt{2}{x}^2+7x+5\sqrt{2}=0$
   4. (iv)$2x^2-x+\frac{1}{8}=0$
   5. (v)$100x^2-20x+1=0$
   Show answer

   (i) $x=5,-2$; (ii) $x=\frac{3}{2},-2$; (iii) $x=-\frac{5\sqrt{2}}{2},-\sqrt{2}$; (iv) $x=\frac{1}{4}$ (equal roots); (v) $x=\frac{1}{10}$ (equal roots).

   Step-by-step solution →
4. Q4 NCERT · Exercise 4.2
   Solve the problems given in Example 1. ​[Example 1:

   1. (i)John and ​Jivanti together have 45 marbles. Both of them lost ​5 marbles each, and the product of the number of marbles they now have is 124. Find how many marbles they had to start with.
   2. (ii)A cottage ​industry produces a certain number of toys in a day. The cost of production of each ​toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ₹ 750. Find ​the number of toys ​produced on that day.]
   Show answer

   (i) The marbles satisfy $x^2-45x+324=0$, so $x=36$ or $x=9$ — John and Jivanti started with 36 and 9 marbles. (ii) The toys satisfy $x^2-55x+750=0$, so $x=25$ or $x=30$ — either number of toys is possible.

   Step-by-step solution →
5. Q5 NCERT · Exercise 4.2
   Find ​two numbers ​whose sum ​is ​27 ​and product ​is 182.
   Show answer

   The two numbers are $13$ and $14$.

   Step-by-step solution →
6. Q6 NCERT · Exercise 4.2
   Find ​two consecutive positive ​integers, sum of ​whose ​squares ​is ​365.
   Show answer

   The two consecutive positive integers are $13$ and $14$.

   Step-by-step solution →
7. Q7 NCERT · Exercise 4.2
   The altitude of ​a right triangle is $7$ cm ​less than ​its base. If the ​hypotenuse is $13$ ​cm, ​find the other two sides.
   Show answer

   Base $=12$ cm and altitude $=5$ cm.

   Step-by-step solution →
8. Q8 NCERT · Exercise 4.2
   A ​cottage ​industry produces ​a ​certain number of pottery articles in ​a day. It was observed on a particular day that the cost of production of each article (in rupees) was $3$ more than twice the number of ​articles produced on that day. If the total cost ​of production on that day was Rs. $90$, find ​the number of articles produced and the cost of each article.
   Show answer

   Number of articles $=6$ and cost of each article $=$ Rs. $15$.

   Step-by-step solution →
9. Q9 Exemplar · Exercise 4.1
   Which of ​the ​following ​is a ​quadratic ​equation?

   1. (a)$x^2+2x+1=(4-x{)}^2+3$
   2. (b)$-2x^2=(5-x)\left(2x-\frac{2}{5}\right)$
   3. (c)$(k+1)x^2+\frac{3}{2}x=7$, where $k=-1$
   4. (d)$x^3-x^2=(x-1{)}^3$
   Show answer

   (d) $x^3-x^2=(x-1{)}^3$

   Step-by-step solution →
10. Q10 Exemplar · Exercise 4.1
    Which ​of ​the following is not ​a quadratic equation?

    1. (a)$2(x-1{)}^2=4x^2-2x+1$
    2. (b)$2x-x^2=x^2+5$
    3. (c)$(\sqrt{2}x+\sqrt{3}{)}^2+x^2=3x^2-5x$
    4. (d)$(x^2+2x{)}^2=x^4+3+4x^3$
    Show answer

    (c) $(\sqrt{2}x+\sqrt{3}{)}^2+x^2=3x^2-5x$

    Step-by-step solution →
11. Q11 Exemplar · Exercise 4.1
    Which of ​the following ​equations has $2$ ​as ​a ​root?

    1. (a)$x^2-4x+5=0$
    2. (b)$x^2+3x-12=0$
    3. (c)$2x^2-7x+6=0$
    4. (d)$3x^2-6x-2=0$
    Show answer

    (c) $2x^2-7x+6=0$

    Step-by-step solution →
12. Q12 Exemplar · Exercise 4.1
    If $\frac{1}{2}$ is ​a root of the ​equation $x^2+kx-\frac{5}{4}=0$, ​then the ​value of $k$ ​is

    1. (a)2
    2. (b)$-2$
    3. (c)$\frac{1}{4}$
    4. (d)$\frac{1}{2}$
    Show answer

    (a) 2

    Step-by-step solution →
13. Q13 Exemplar · Exercise 4.1
    Which of the ​following ​equations has ​the ​sum ​of its roots ​as 3?

    1. (a)$2x^2-3x+6=0$
    2. (b)$-x^2+3x-3=0$
    3. (c)$\sqrt{2}{x}^2-\frac{3}{\sqrt{2}}x+1=0$
    4. (d)$3x^2-3x+3=0$
    Show answer

    (b) $-x^2+3x-3=0$

    Step-by-step solution →
14. Q14 Exemplar · Exercise 4.1
    Values ​of $k$ ​for ​which the quadratic ​equation $2x^2-kx+k=0$ has ​equal ​roots is

    1. (a)$0$ only
    2. (b)$4$
    3. (c)$8$ only
    4. (d)$0,8$
    Show answer

    (d) $0,8$

    Step-by-step solution →
15. Q15 Exemplar · Exercise 4.1
    Which constant ​must ​be added ​and subtracted ​to ​solve the quadratic ​equation $9x^2+\frac{3}{4}x-\sqrt{2}=0$ by ​the ​method of completing the square?

    1. (a)$\frac{1}{8}$
    2. (b)$\frac{1}{64}$
    3. (c)$\frac{1}{4}$
    4. (d)$\frac{9}{64}$
    Show answer

    (b) $\frac{1}{64}$

    Step-by-step solution →
16. Q16 Exemplar · Exercise 4.1
    The ​quadratic ​equation $2x^2-\sqrt{5}x+1=0$ ​has

    1. (a)two distinct real roots
    2. (b)two equal real roots
    3. (c)no real roots
    4. (d)more than 2 real roots
    Show answer

    (c) no real roots

    Step-by-step solution →
17. Q17 Exemplar · Exercise 4.1
    Which ​of ​the ​following equations has ​two distinct ​real ​roots?

    1. (a)$2x^2-3\sqrt{2}x+\frac{9}{4}=0$
    2. (b)$x^2+x-5=0$
    3. (c)$x^2+3x+2\sqrt{2}=0$
    4. (d)$5x^2-3x+1=0$
    Show answer

    (b) $x^2+x-5=0$

    Step-by-step solution →
18. Q18 Exemplar · Exercise 4.1
    Which ​of the following equations ​has no ​real ​roots?

    1. (a)$x^2-4x+3\sqrt{2}=0$
    2. (b)$x^2+4x-3\sqrt{2}=0$
    3. (c)$x^2-4x-3\sqrt{2}=0$
    4. (d)$3x^2+4\sqrt{3}x+4=0$
    Show answer

    (a) $x^2-4x+3\sqrt{2}=0$

    Step-by-step solution →
19. Q19 Exemplar · Exercise 4.1
    $(x^2+1{)}^2-x^2=0$ has

    1. (a)four real roots
    2. (b)two real roots
    3. (c)no real roots
    4. (d)one real root
    Show answer

    (c) no real roots

    Step-by-step solution →
20. Q20 Exemplar · Exercise 4.2
    State ​whether the ​following ​quadratic equations have two ​distinct real ​roots. Justify ​your ​answer.

    1. (i)$x^2-3x+4=0$
    2. (ii)$2x^2+x-1=0$
    3. (iii)$2x^2-6x+\frac{9}{2}=0$
    4. (iv)$3x^2-4x+1=0$
    5. (v)$(x+4{)}^2-8x=0$
    6. (vi)$(x-\sqrt{2}{)}^2-2(x+1)=0$
    7. (vii)$\sqrt{2}{x}^2-\frac{3}{\sqrt{2}}x+\frac{1}{\sqrt{2}}=0$
    8. (viii)$x(1-x)-2=0$
    9. (ix)$(x-1)(x+2)+2=0$
    10. (x)$(x+1)(x-2)+x=0$
    Show answer

    Two distinct real roots: (ii), (iv), (vi), (vii), (ix), (x). No real roots: (i), (v), (viii). Equal (not distinct) roots: (iii).

    Step-by-step solution →