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.
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 ›
Q4.
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 ›
Q5.
If $α$ and $β$ are the roots of the equation $2 x^{2} + 2 p x + p^{2} = 0$ , where p is a non-zero real number, and $α^{4}$ and $β^{4}$ are the roots of $x^{2} - r x + s = 0$ , then the roots of $2 x^{2} - 4 p^{2} x + 4 p^{4} - 2 r = 0$ are:
A. Real and unequalB. Equal and zeroC. ImaginaryD. Equal and non-zero
hardAsked in NIMCET 2014View solution ›
Q6.
Let $α$ and $β$ be the roots of the equation $x^{2} - p x + r = 0$ , and let $\frac{α}{2}$ and $2 β$ be the roots of the equation $x^{2} - q x + r = 0$ . Then the value of r is
A. $\frac{2}{9} (p - q) (2 q - p)$ B. $\frac{2}{9} (q - p) (2 q - p)$ C. $\frac{2}{9} (q - 2 p) (2 q - p)$ D. $\frac{2}{9} (2 p - q) (2 q - p)$
hardAsked in NIMCET 2018View solution ›
Q7.
If $α$ and $β$ are the two roots of the quadratic equation $x^{2} + a x + b = 0$ $(ab \neq = 0)$ , then the quadratic equation whose roots are $\frac{1}{α ^{3} + α}$ and $\frac{1}{β ^{3} + β}$ is
A. $b (b^{2} + 1 + a^{2} + 2 b) x^{2} - (a^{3} + a - 3 ab) x + 1 = 0$ B. $b (b^{2} + 1 + a^{2} - 2 b) x^{2} - (a^{3} + a - 3 ab) x + 1 = 0$ C. $b (b^{2} + 1 + a^{2} + 2 b) x^{2} + (a^{3} - a - 3 ab) x + 1 = 0$ D. $b (b^{2} + 1 + a^{2} - 2 b) x^{2} + (a^{3} + a - 3 ab) x + 1 = 0$
hardAsked in NIMCET 2025View solution ›