Quadratic Equations — Formula Sheet

ax^2 + bx + c = 0, a ≠ 0

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

Factorisation method: If (p)(q) = 0, then p = 0 or q = 0

Repeated root case: x = -\frac{b}{2a}

Discriminant: b^2 - 4ac

If b^2 - 4ac > 0, two distinct real roots

If b^2 - 4ac = 0, two equal real roots

If b^2 - 4ac < 0, no real roots

Area of rectangle = length × breadth

Pythagoras theorem: AP^2 + BP^2 = AB^2

John's marbles problem leads to x^2 - 45x + 324 = 0.

The toys production problem leads to x^2 - 55x + 750 = 0.

Babylonians solved equations like x^2 - px + q = 0.

Brahmagupta gave a formula for ax^2 + bx = c.

A real number α is a root of ax^2 + bx + c = 0 if aα^2 + bα + c = 0.

Quadratic equations are solved by factorisation, completing the square, and the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a.