Some quadratics factorise neatly, but many do not. The quadratic formula solves any quadratic written in the standard form ax^2 + bx + c = 0 (with a \neq 0) — you just read off a, b and c and substitute:

The \pm sign is doing two jobs at once: take the + for one solution and the - for the other. It comes straight from completing the square on the general equation — that is the formula's proof. Where the neat shortcut of factorising fails, this formula always works.

See it worked through

Watch x^2 + 5x + 6 = 0 solved one line at a time: name the coefficients, substitute, compute the discriminant b^2 - 4ac, then split the \pm into the two roots. Step through it.

How many solutions? The discriminant

The part under the square root, b^2 - 4ac, is called the discriminant. Without solving the whole equation, its sign tells you how many real solutions the quadratic has:

• b^2 - 4ac > 0 — the root is a real number added and subtracted, giving two distinct real solutions.
• b^2 - 4ac = 0 — the \pm\sqrt{0} adds nothing, so both solutions collapse into one (a repeated root).
• b^2 - 4ac < 0 — you would need the square root of a negative number, which is not real, so there are no real solutions.

See it explained

Sal Khan substitutes the coefficients into the quadratic formula and simplifies to the two roots.