The quadratic formula solves any equation of the form a x squared plus b x plus c equals 0, straight from its coefficients. Learn the step-by-step method, how the discriminant b squared minus 4 a c tells you the number of real solutions, and a fully worked example.
What the quadratic formula is
The quadratic formula solves any quadratic equation written in the standard form ax² + bx + c = 0. It gives both solutions directly from the coefficients a, b, and c:
The quadratic formula, with the discriminant b² − 4ac highlighted.
Unlike solving by factoring, the formula works even when the equation does not factor neatly, so it is the reliable fallback for every quadratic.
How to use it, step by step
Write the equation as ax² + bx + c = 0 and read off a, b, and c.
Substitute them into x = (−b ± √(b² − 4ac)) / 2a.
Simplify under the square root first, then compute the two values (one with +, one with −).
The discriminant
The expression under the square root, b² − 4ac, is the discriminant. It tells you how many real solutions to expect before you finish: positive gives two, zero gives one, negative gives none (two complex ones). That is the whole idea behind the nature of the roots.
Worked example
Solve x² − 5x + 6 = 0. Here a = 1, b = −5, c = 6. Discriminant: (−5)² − 4×1×6 = 25 − 24 = 1. So x = (5 ± √1) / 2 = (5 ± 1) / 2, giving x = 3 or x = 2. When a formula feels heavy, completing the square reaches the same answers.