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.
Let A be a real number. Then the roots of the equation $x^{2} - 4 x - l o g_{2} A = 0$ are real and distinct if and only if
A. $A < \frac{1}{16}$ B. $A > \frac{1}{8}$ C. $A > \frac{1}{16}$ D. $A < \frac{1}{8}$
hardAsked in CAT 2019View solution ›
Q2.
The number of distinct real roots of the equation $(x + \frac{1}{x})^{2} - 3 (x + \frac{1}{x}) + 2 = 0$ is
A. 0B. 1C. 2D. 4
hardAsked in CAT 2020View solution ›
Q3.
The roots of $2 x^{2} - 7 x + 3 = 0$ are:
A. $1, 3$ B. $\frac{1}{2}, 3$ C. $- \frac{1}{2}, 3$ D. $\frac{1}{2}, - 3$
easyView solution ›
Q4.
For what value of $k$ does $x^{2} - 6 x + k = 0$ have equal real roots?
A. $6$ B. $12$ C. $9$ D. $36$
easyView solution ›
Q5.
The equation $x^{2} + 4 x + k = 0$ has two distinct real roots when:
A. $k < 4$ B. $k = 4$ C. $k > 4$ D. $k \geq 4$
easyView solution ›
Q6.
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 ›
Q7.
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 ›
Q8.
The roots of $x^{2} - 10 x + k = 0$ are in the ratio $2 : 3$ . Find $k$ .
A. $24$ B. $20$ C. $25$ D. $30$
mediumView solution ›
Q9.
The solutions of $x^{2} - 2 x - 2 = 0$ are:
A. $- 1 \pm 3$ B. $1 \pm 2$ C. $1 \pm 3$ D. $2 \pm 3$
mediumView solution ›
Q10.
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. $\frac{1}{2}$
mediumView solution ›
Q11.
A rectangle has area 40 square units. Its length is 3 units more than its width. Find its width in units.
A. $5$ B. $8$ C. $4$ D. $10$
easyView solution ›
Q12.
If $α$ and $β$ are the roots of $x^{2} - 5 x + 6 = 0$ , find $\frac{1}{α} + \frac{1}{β}$ .
A. $\frac{6}{5}$ B. $\frac{5}{6}$ C. $\frac{1}{6}$ D. $5$
easyView solution ›
Q13.
Let $a, b, c$ be non-zero real numbers such that $a + b + c \neq = 0$ and $4 a - 2 b + c \neq = 0$ . If $α$ and $β$ are the roots of $a x^{2} + b x + c = 0$ , then which of the following equations has the roots $\frac{α + 2}{α - 1}$ and $\frac{β + 2}{β - 1}$ ?
A. $(a + b + c) x^{2} - (4 a + b - 2 c) x + (4 a - 2 b + c) = 0$ B. $(a + b + c) x^{2} + (4 a - 2 b + c) x + (4 a + b - 2 c) = 0$ C. $(a + b + c) x^{2} - (4 a - 2 b + c) x + (4 a + b - 2 c) = 0$ D. $(a + b + c) x^{2} + (4 a + b - 2 c) x + (4 a - 2 b + c) = 0$
hardAsked in NIMCET 2026View solution ›
Q14.
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 ›
Q15.
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 ›
Q16.
The equation $(cos p - 1) x^{2} + (cos p) x + s in p = 0$ , where $x$ is a variable, has real roots. Then the interval of $p$ is
A. $(0, 2 π)$ B. $(- π, 0)$ C. $(- \frac{π}{2}, \frac{π}{2})$ D. $(0, π)$
hardAsked in NIMCET 2012View solution ›
Q17.
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 ›
Q18.
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 ›
Q19.
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 ›
Q20.
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 ›

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