For which values of k does 3x² + 2x + k = 0 have two distinct real roots?

For which values of $k$ does $3 x^{2} + 2 x + k = 0$ have two distinct real roots?

Class 10Quadratic EquationsAlgebra — Quadratics

Two distinct real roots need

D > 0 : 2^{2} - 4 (3) k > 0

→

4 - 12 k > 0

→

4 > 12 k

→

k < \frac{1}{3}

. Dividing by 12 and forgetting that the inequality flips when you move a negative term across gives

k > \frac{1}{3}

;

k = \frac{1}{3}

is exactly the equal-roots case

(D = 0)

, which is excluded;

k < 3

comes from dividing 4 by 12 upside down.

Answer: C)

k < \frac{1}{3}

D = b^{2} - 4 a c

classifies the roots of

a x^{2} + b x + c = 0 : - * * D > 0

**: TWO distinct real roots. Parabola crosses

x

-axis at two points.
- **

D = 0

**: ONE repeated real root (called a 'double root' or 'two equal roots'). Parabola touches

x

-axis at the vertex.
- **

D < 0

**: NO real roots. Parabola does not touch the

x

-axis at all.

**Use case**: 'For what values of

k

does

k x^{2} + 2 x + 1 = 0

have equal roots?' → set

D = 0 : 4 - 4 k = 0

→

k = 1

.

You can answer 'how many real roots?' WITHOUT computing them — just compute

D

Don't just memorise the answer — build a daily practice habit.