Class 10 · Chapter 4

Quadratic Equations

Standard form, factorisation, completing the square, quadratic formula, nature of roots and word problems.

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Mathematics · Class 10 (2026-27)

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Detailed theory

A quadratic equation is of the form

ax^2 + bx + c = 0

with

a \ne 0

. A number

\alpha

is a root (or zero) if

a\alpha^2 + b\alpha + c = 0

. Every quadratic has at most two roots. Real coefficients need not yield real roots — the discriminant decides.

Key concepts & formulas

Standard form:

ax^2 + bx + c = 0,\; a\ne 0

Quadratic formula:

x = \dfrac{-b\pm\sqrt{b^2 - 4ac}}{2a}

Discriminant

D = b^2 - 4ac

>0

two distinct real;

=0

equal;

<0

no real.

Sum

= -b/a

, product

= c/a

(same as polynomials).

Completing the square rewrites as

(x + p)^2 = q

Always convert to standard form before applying formulae.