How many distinct real roots does 100x² − 20x + 1 = 0 have, and what are they?

How many distinct real roots does $100 x^{2} - 20 x + 1 = 0$ have, and what are they?

Class 10Quadratic EquationsAlgebra — Quadratics

100 x^{2} - 20 x + 1

is a perfect square:

(10 x - 1)^{2} = 0

. So there is a single distinct root,

x = \frac{1}{10}

, occurring twice. Check:

100 (0.01) - 20 (0.1) + 1 = 1 - 2 + 1 = 0

. A perfect square never produces a ± pair, so the first option is wrong;

x = 10

inverts the fraction; and

D = 400 - 400 = 0

(not negative), so real roots do exist.

Answer: B) One repeated root,

x = \frac{1}{10}

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.