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 2 of 4

1. Q21 Exemplar · Exercise 4.2
   Write whether the following ​statements are true or false. Justify your answers.

   1. (i)Every quadratic equation has exactly one root.
   2. (ii)Every quadratic equation has at least one real root.
   3. (iii)Every quadratic equation has at least two roots.
   4. (iv)Every quadratic equation ​has at most ​two roots.
   5. (v)If the coefficient of $x^2$ ​and the constant term ​of a quadratic equation have opposite ​signs, then the quadratic equation has real roots.
   6. (vi)If the coefficient of $x^2$ and the constant term have the same sign and if the coefficient of $x$ ​term ​is zero, then the quadratic equation has no real roots.
   Show answer

   (i) False -- it has two roots; (ii) False -- e.g. $x^2+1=0$ has no real root; (iii) False -- exactly two, not 'at least two' more; (iv) True; (v) True since $D=b^2-4ac>0$ when $ac<0$; (vi) True since $D=-4ac<0$ when $ac>0,b=0$.

   Step-by-step solution →
2. Q22 Exemplar · Exercise 4.2
   A ​quadratic equation with ​integral ​coefficient ​has integral ​roots. Justify ​your ​answer.
   Show answer

   No, it is not true. For example $x^2-x-4=0$ has integral coefficients but roots $\frac{1\pm \sqrt{17}}{2}$, which are not integers.

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3. Q23 Exemplar · Exercise 4.2
   Does ​there exist ​a ​quadratic ​equation whose coefficients are rational but ​both of its ​roots are irrational? ​Justify your answer.
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   Yes. For example $x^2-4x+1=0$ has rational coefficients but irrational roots $2\pm \sqrt{3}$.

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4. Q24 Exemplar · Exercise 4.2
   Does ​there exist ​a ​quadratic equation whose coefficients are ​all distinct irrationals ​but both the ​roots are ​rationals? Why?
   Show answer

   Yes. For example $\sqrt{2}{x}^2-3\sqrt{2}x+2\sqrt{2}=0$ has all-distinct-irrational coefficients but rational roots $1$ and $2$.

   Step-by-step solution →
5. Q25 Exemplar · Exercise 4.2
   Is $0.2$ ​a ​root ​of ​the equation $x^2-0.4=0$? Justify.
   Show answer

   No, $0.2$ is not a root: $p(0.2)=0.04-0.4=-0.36\ne 0$. The actual roots are $x=\pm \sqrt{0.4}\approx \pm 0.632$.

   Step-by-step solution →
6. Q26 Exemplar · Exercise 4.2
   If $b=0$, $c<0$, ​is ​it true that ​the ​roots of $x^2+bx+c=0$ are numerically ​equal and ​opposite in ​sign? ​Justify.
   Show answer

   Yes, it is true. With $b=0$ the equation is $x^2=-c$; since $c<0$, $-c>0$, so the roots are $+\sqrt{-c}$ and $-\sqrt{-c}$ - numerically equal and opposite in sign.

   Step-by-step solution →
7. Q27 Exemplar · Exercise 4.3
   Find the ​roots of the quadratic ​equations by using the ​quadratic formula ​in each of the ​following:

   1. (i)$2x^2-3x-5=0$
   2. (ii)$5x^2+13x+8=0$
   3. (iii)$-3x^2+5x+12=0$
   4. (iv)$-x^2+7x-10=0$ ​
   5. (v)$x^2+2\sqrt{2}x-6=0$
   6. (vi)$x^2-3\sqrt{5}x+10=0$ ​
   7. (vii)$(1/2)x^2-\sqrt{11}x+1=0$.
   Show answer

   (i) $x=5/2,-1$; (ii) $x=-1,-8/5$; (iii) $x=3,-4/3$; (iv) $x=5,2$; (v) $x=\sqrt{2},-3\sqrt{2}$; (vi) $x=2\sqrt{5},\sqrt{5}$; (vii) $x=\sqrt{11}+3,\sqrt{11}-3$.

   Step-by-step solution →
8. Q28 Exemplar · Exercise 4.3
   Find the roots ​of ​the following quadratic ​equations by ​the ​factorisation ​method:

   1. (i)$2x^2+\frac{5}{3}x-2=0$
   2. (ii)$\frac{2}{5}{x}^2-x-\frac{3}{5}=0$
   3. (iii)$3\sqrt{2}{x}^2-5x-\sqrt{2}=0$ ​
   4. (iv)$3x^2+5\sqrt{5}x-10=0$
   5. (v)$21x^2-2x+\frac{1}{21}=0$
   Show answer

   (i) $x=\frac{2}{3},-\frac{3}{2}$; (ii) $x=3,-\frac{1}{2}$; (iii) $x=\sqrt{2},-\frac{\sqrt{2}}{6}$; (iv) $x=\frac{\sqrt{5}}{3},-2\sqrt{5}$; (v) $x=\frac{1}{21}$ (repeated).

   Step-by-step solution →
9. Q29 CBSE 2026
   The roots ​of the ​quadratic ​equation $(x-1{)}^2=16$ ​are ​:

   1. (a)$5,3$
   2. (b)$4,-4$
   3. (c)$5,-3$
   4. (d)$-5,3$
   Show answer

   (c) $5,-3$

   Step-by-step solution →
10. Q30 CBSE 2026
    If ​the ​quadratic equation $9x^2+8kx+16=0$ ​has real ​and ​equal roots, ​then the ​value of $k$ ​is

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

    (a) $3$

    (b) $-3$

    Step-by-step solution →
11. Q31 CBSE 2026
    The ​value of $k$ for ​which the equation $k{x}^2-6x-4=0$ ​has real ​and ​equal ​roots, is

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

    (c) $-\frac{9}{4}$

    Step-by-step solution →
12. Q32 CBSE 2026
    The roots ​of the ​quadratic ​equation $x^2+9=0$ ​are

    1. (a)real and equal
    2. (b)not real
    3. (c)real and negative of each other
    4. (d)rational numbers
    Show answer

    (b) not real

    Step-by-step solution →
13. Q33 CBSE 2026
    The ​value ​of $k$ ​for which the equation $2k{x}^2-6x+3=0$ ​has ​real ​and equal ​roots, is :

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

    (a) $\frac{3}{2}$

    Step-by-step solution →
14. Q34 CBSE 2026
    If $x=\sqrt{3}$ is ​a ​solution of ​the ​equation $a{x}^2+\sqrt{3}x-12=0$, ​then

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

    (a) $a=3$

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15. Q35 CBSE 2026
    If the equation $q{x}^2+px-r=0$ $(q\ne 0)$ ​has ​real ​and equal roots, then which of the following is ​true ​?

    1. (a)$p^2=qr$
    2. (b)$p^2=-4qr$
    3. (c)$q^2=4pr$
    4. (d)$q^2=-4pr$
    Show answer

    (b) $p^2=-4qr$

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16. Q36 CBSE 2026
    The ​value ​of ​'$m$' ​for which ​the quadratic ​equation $x^2-2x+m=0$ has real and ​equal roots is :

    1. (a)$-1$
    2. (b)$0$
    3. (c)$4$
    4. (d)$1$
    Show answer

    (d) $1$

    Step-by-step solution →
17. Q37 CBSE 2026
    Which of the ​equations ​among the ​following ​is/are quadratic equation(s) ​? $q_1:x^2+x=(x+1{)}^2$, $q_2:x-1=x^2-1$, $q_3:x^4=x^2$, $q_4:\sqrt{x}=x^2\sqrt{x}+1$

    1. (a)$q_1$ only
    2. (b)$q_1,q_2$ and $q_3$ only
    3. (c)$q_2$ only
    4. (d)$q_2$ and $q_4$ only
    Show answer

    (c) $q_2$ only

    Step-by-step solution →
18. Q38 CBSE 2026
    If $x=-1$ is a ​root of ​the equation $a{x}^2-bx+3=0$, ​then ​:

    1. (a)$-a+b-3=0$
    2. (b)$a-b-3=0$
    3. (c)$-a-b+3=0$
    4. (d)$a+b+3=0$
    Show answer

    (d) $a+b+3=0$

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19. Q39 CBSE 2026
    Which of the following ​statements is ​true ​for ​the quadratic ​equation $a{x}^2+x+a=0$ ($a\ne 0$) ​?

    1. (a)The roots of the given equation are reciprocal of each other.
    2. (b)The roots of the given equation are always real.
    3. (c)The roots of the given equation are always positive.
    4. (d)The sum of roots of the given equation is $0$.
    Show answer

    (a) The roots of the given equation are reciprocal of each other.

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20. Q40 CBSE 2026
    The ​discriminant ​of the quadratic ​equation $a{x}^2+x+a=0$ ​is ​:

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

    (b) $1-4a^2$

    Step-by-step solution →