The Quadratic Formula and the Discriminant

For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. It works when factoring fails.

Standard form first

Move everything to one side before reading a, b and c. Reading them from an unarranged equation is the usual mistake.

The discriminant

The part under the root, b² − 4ac, is the discriminant. Its sign alone tells you what kind of solutions to expect.

Three cases

Positive gives two real roots. Zero gives one repeated root. Negative gives two complex roots, which are conjugates of each other.

What it looks like

The three cases are a parabola crossing the x-axis twice, touching it once, or missing it entirely.

A bonus

If the discriminant is a perfect square, the roots are rational and the quadratic would have factored.

The formula is completing the square, done once

Completing the square on ax² + bx + c = 0 in general produces x = (−b ± √(b² − 4ac))/(2a). The formula exists so nobody has to repeat that derivation, but doing it once makes the formula impossible to misremember.

The discriminant predicts the answer

Positive gives two distinct real roots, zero gives one repeated root, negative gives two complex conjugate roots. Computing b² − 4ac first tells you what kind of answer to expect before any square roots are taken.

A perfect-square discriminant means it factors

If b² − 4ac is a perfect square, the quadratic factors over the rationals and factoring would have been quicker. Checking the discriminant is a fast way to decide which method to use.

Where it goes wrong

Forgetting that the whole numerator is divided by 2a, and losing the minus on −b, are the two standard errors. Writing the formula out fully before substituting, rather than substituting from memory, prevents both.