Quadratic Equations Questions with Solutions

34 questions6 easy21 medium7 hard

A quadratic equation has the form ax^2 + bx + c = 0, where a is not zero. This topic covers solving quadratics by factorisation and by the quadratic formula, the nature of roots using the discriminant b^2 - 4ac, the sum (-b/a) and product (c/a) of roots, and forming an equation from its roots. In aptitude exams, a common question type gives two quadratic equations and asks you to compare their roots (whether x > y, x < y and so on). In engineering and MCA entrance exams, questions also test conditions on roots, common roots, and symmetric expressions such as the sum of squares of roots. Using the sum and product of roots is usually far faster than finding the roots themselves. Practise the quadratic equation questions below, each with a step-by-step solution.

All Questions Easy Medium Hard

Q1.
If $α$ and $β$ are the roots of $3 x^{2} - 5 x - 2 = 0$ , find $α^{2} + β^{2}$ .
A. $\frac{13}{9}$ B. $\frac{25}{9}$ C. $\frac{31}{9}$ D. $\frac{37}{9}$
mediumView solution ›
Q2.
An equation whose roots are the squares of the roots of $x^{2} - 5 x + 6 = 0$ is:
A. $x^{2} - 25 x + 36 = 0$ B. $x^{2} - 13 x + 36 = 0$ C. $x^{2} - 13 x + 6 = 0$ D. $x^{2} + 13 x + 36 = 0$
mediumView solution ›
Q3.
The roots of $x^{2} - 10 x + k = 0$ are in the ratio $2 : 3$ . Find $k$ .
A. $24$ B. $20$ C. $25$ D. $30$
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Q4.
The solutions of $x^{2} - 2 x - 2 = 0$ are:
A. $- 1 \pm 3$ B. $1 \pm 2$ C. $1 \pm 3$ D. $2 \pm 3$
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Q5.
If $2$ is a root of $2 x^{2} - 5 x + k = 0$ , find the other root.
A. $- \frac{1}{2}$ B. $\frac{3}{2}$ C. $3$ D. $\frac{1}{2}$
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Q6.
If the equation $x^{2} - 6 x + 8 = a$ has four real solutions, then find the value of $a$ .
A. $a = 0$ B. $a = 1$ C. $a \in (0, 1)$ D. $a \in [1, 2]$
mediumAsked in NIMCET 2023View solution ›
Q7.
Let $f (x)$ be a polynomial function of second degree and $f (1) = f (- 1)$ . If a, b, c are in A.P., then $f' (a)$ , $f' (b)$ , $f' (c)$ are in
A. GPB. HPC. AGPD. AP
mediumAsked in NIMCET 2013View solution ›
Q8.
The least integral value of $k$ for which $(k - 2) x^{2} + 8 x + k + 4 > 0$ for all real $x$ is
A. $5$ B. $4$ C. $3$ D. $6$
mediumAsked in NIMCET 2011View solution ›
Q9.
If $2 x^{4} + x^{3} - 11 x^{2} + x + 2 = 0$ , then the values of $x + \frac{1}{x}$ are:
A. $- 3, \frac{5}{2}$ B. $- \frac{1}{5}, 3$ C. $\frac{2}{5}, \frac{1}{3}$ D. $\frac{1}{3}, - 5$
mediumAsked in NIMCET 2009View solution ›
Q10.
If $a, b$ are the roots of $x^{2} + p x + 1 = 0$ and $c, d$ are the roots of $x^{2} + q x + 1 = 0$ , then the value of $E = (a - c) (b - c) (a + d) (b + d)$ is
A. $p^{2} - q^{2}$ B. $q^{2} - p^{2}$ C. $q^{2} + p^{2}$ D. None of these
mediumAsked in NIMCET 2008View solution ›
Q11.
The number of roots of the equation $x^{2} - x - 6 = x + 2$ is
A. $2$ B. $3$ C. $4$ D. None of these
mediumAsked in NIMCET 2008View solution ›
Q12.
Roots of equation $a x^{2} - 2 b x + c = 0$ are n and m, then the value of $\frac{b}{a n ^{2} + c} + \frac{b}{a m ^{2} + c}$ is
A. $\frac{c}{a}$ B. $\frac{b}{a}$ C. $\frac{a}{c}$ D. $\frac{b}{c}$
mediumAsked in NIMCET 2020View solution ›
Q13.
If $α \neq = β$ and $α^{2} = 5 α - 3$ , $β^{2} = 5 β - 3$ , then the equation whose roots are $\frac{α}{β}$ and $\frac{β}{α}$ is
A. $3 x^{2} - 25 x + 3 = 0$ B. $3 x^{2} + 5 x + 3 = 0$ C. $3 x^{2} - 5 x + 3 = 0$ D. $3 x^{2} - 19 x + 3 = 0$
mediumAsked in NIMCET 2021View solution ›
Q14.
If $α, β$ are the roots of $x^{2} - x - 1 = 0$ and $A_{n} = α^{n} + β^{n}$ , then the arithmetic mean of $A_{n - 1}$ and $A_{n}$ is
A. $2 A_{n} - 1$ B. $\frac{1}{2} A_{n + 1}$ C. $2 A_{n} - 2$ D. None of the above
mediumAsked in NIMCET 2022View solution ›
Q15.
If the roots of the quadratic equation $x^{2} + p x + q = 0$ are $t an 30^{\circ}$ and $t an 15^{\circ}$ respectively, then the value of $2 + p - q$ is
A. 3B. 0C. 1D. 2
mediumAsked in NIMCET 2022View solution ›
Q16.
The value of $k$ for which the equation $(k - 2) x^{2} + 8 x + k + 4 = 0$ has both roots real, distinct and negative is
A. 0B. 2C. 3D. -4
mediumAsked in NIMCET 2015View solution ›
Q17.
If $α$ , $β$ are the roots of the equation $x^{2} - 2 x cos θ + 1 = 0$ , then the equation having roots $α^{n}$ and $β^{n}$ is
A. $x^{2} - (2 cos n θ) x + 1 = 0$ B. $x^{2} - (2 cos n θ) x - 1 = 0$ C. $x^{2} + (2 cos n θ) x + 1 = 0$ D. $x^{2} + (2 cos n θ) x - 1 = 0$
mediumAsked in NIMCET 2017View solution ›
Q18.
The equation $(x - a)^{3} + (x - b)^{3} + (x - c)^{3} = 0$ has
A. All three real rootsB. One real and two imaginary rootsC. Three real roots, namely x = a, x = b, x = cD. None of these
mediumAsked in NIMCET 2017View solution ›
Q19.
The quadratic equation whose roots are $s in^{2} 18^{\circ}$ and $cos^{2} 36^{\circ}$ is
A. $16 x^{2} - 12 x + 1 = 0$ B. $16 x^{2} + 12 x + 1 = 0$ C. $16 x^{2} - 12 x - 1 = 0$ D. None of these
mediumAsked in NIMCET 2018View solution ›
Q20.
If x is real, then the minimum value of $\frac{x ^{2} - x + 1}{x ^{2} + x + 1}$ is:
A. $\frac{1}{2}$ B. 2C. 3D. $\frac{1}{3}$
mediumAsked in NIMCET 2019View solution ›

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