Q: The general form of a quadratic equation is:

ax^2 + bx + c = 0, with a \neq 0. A quadratic has degree 2: ax² + bx + c = 0 with a ≠ 0 (if a = 0 it becomes linear).

Q: If D < 0, the roots of a quadratic (with real coefficients) are:

Complex conjugates (no real roots). A negative discriminant means no real solutions — the roots are complex.

Q: For roots α and β of ax^2 + bx + c = 0, the sum of the roots is:

-\dfrac{b}{a}. α + β = −b/a.

Q: A quadratic equation always has:

Exactly two roots (real or complex). By the fundamental theorem, a degree-2 equation has exactly two roots (counting multiplicity).

Q: If α and β are the roots, the quadratic equation can be written as:

x^2 - (\alpha+\beta)x + \alpha\beta = 0. Equation = x² − (sum)x + (product) = 0.

Q: When a physics kinematics problem gives a quadratic in time t, a negative value of t is:

Rejected as physically meaningless. Time cannot be negative, so the negative root is discarded in physical problems.

Question 1

The general form of a quadratic equation is:

Accepted Answer

ax^2 + bx + c = 0, with a 
eq 0. A quadratic has degree 2: ax² + bx + c = 0 with a ≠ 0 (if a = 0 it becomes linear).

Question 2

The roots of ax^2 + bx + c = 0 are given by the quadratic formula:

Accepted Answer

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}. x = [−b ± √(b² − 4ac)]/2a.

Question 3

The discriminant of a quadratic equation is:

Accepted Answer

D = b^2 - 4ac. The discriminant D = b² − 4ac decides the nature of the roots.

Question 4

If the discriminant D > 0, the roots are:

Accepted Answer

Real and distinct. D > 0 gives two different real roots.

Question 5

If D = 0, the quadratic has roots that are:

Accepted Answer

Real and equal (repeated). D = 0 gives a single repeated real root, x = −b/2a.

Question 6

If D < 0, the roots of a quadratic (with real coefficients) are:

Accepted Answer

Complex conjugates (no real roots). A negative discriminant means no real solutions — the roots are complex.

Question 7

For roots α and β of ax^2 + bx + c = 0, the sum of the roots is:

Accepted Answer

-\dfrac{b}{a}. α + β = −b/a.

Question 8

The product of the roots of ax^2 + bx + c = 0 is:

Accepted Answer

\dfrac{c}{a}. αβ = c/a.

Question 9

A quadratic equation always has:

Accepted Answer

Exactly two roots (real or complex). By the fundamental theorem, a degree-2 equation has exactly two roots (counting multiplicity).

Question 10

If α and β are the roots, the quadratic equation can be written as:

Accepted Answer

x^2 - (\alpha+\beta)x + \alpha\beta = 0. Equation = x² − (sum)x + (product) = 0.

Question 11

When a physics kinematics problem gives a quadratic in time t, a negative value of t is:

Accepted Answer

Rejected as physically meaningless. Time cannot be negative, so the negative root is discarded in physical problems.

Question 12

The condition for the roots of ax^2 + bx + c = 0 to be equal is:

Accepted Answer

b^2 = 4ac. Equal roots require D = 0, i.e. b² = 4ac.

Question 13

If one root of ax^2 + bx + c = 0 is zero, then:

Accepted Answer

c = 0. Product of roots = c/a = 0 ⇒ c = 0.

Question 14

The vertex (turning point) of the parabola y = ax^2 + bx + c occurs at:

Accepted Answer

x = -\dfrac{b}{2a}. The extremum is at x = −b/2a (also midway between the two roots).

Question 15

The roots of x^2 - 5x + 6 = 0 are:

Accepted Answer

2 and 3. Factor: (x−2)(x−3) = 0 ⇒ x = 2, 3.

Question 16

The roots of x^2 - 7x + 12 = 0 are:

Accepted Answer

3 and 4. (x−3)(x−4) = 0 ⇒ x = 3, 4.

Question 17

The roots of x^2 - 4 = 0 are:

Accepted Answer

+2 and −2. x² = 4 ⇒ x = ±2.

Question 18

The roots of x^2 + 2x - 3 = 0 are:

Accepted Answer

1 and −3. (x−1)(x+3) = 0 ⇒ x = 1, −3.

Question 19

The roots of 2x^2 - 5x + 2 = 0 are:

Accepted Answer

2 and 1/2. x = [5 ± √(25−16)]/4 = (5 ± 3)/4 ⇒ 2 or 1/2.

Question 20

The roots of x^2 - 6x + 9 = 0 are:

Accepted Answer

3 (repeated). (x−3)² = 0 ⇒ x = 3 (equal roots, D = 0).

Roots, Nature & Physics Applications — JEE Main Basic Maths MCQs with Solutions

Questions with solutions