Quadratic Equation

A quadratic equation has the standard form

$ax^2 + bx + c = 0, \quad a \neq 0.$

The graph of $y = ax^2 + bx + c$ is a parabola; the equation's solutions are the x-values where the parabola crosses the x-axis.

Quadratics can have 0, 1, or 2 real solutions, governed by the discriminant $\Delta = b^2 - 4ac$ (positive ⇒ two real roots; zero ⇒ one repeated root; negative ⇒ two complex conjugate roots).

Standard solving techniques include the quadratic formula $x = \frac{-b \pm \sqrt{\Delta}}{2a}$, factoring, and completing the square. Quadratics appear throughout science: projectile motion, optimisation, parabolic mirrors, and even the simplest quantum-mechanical models.