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Using quadratic formula to solve quadratic equations

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The Quadratic Formula

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 and the discriminant The quadratic formula: x equals negative b, plus or minus the square root of (b squared minus 4 a c), all divided by 2 a. The part under the square root, b squared minus 4 a c, is the discriminant, which tells you how many real solutions the equation has. x = −b ± √(b² − 4ac) 2a discriminant: b² − 4ac
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.

  1. Write the equation as ax² + bx + c = 0 and read off a, b, and c.
  2. Substitute them into x = (−b ± √(b² − 4ac)) / 2a.
  3. Simplify under the square root first, then compute the two values (one with +, one with −).

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.

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.

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