The Quadratic Formula

The quadratic formula solves every quadratic, factorable or not.

For ax² + bx + c = 0:
x = (−b ± √(b² − 4ac)) / (2a)

Write the equation in standard form first and identify a, b, c *with their signs*. In 2x² − 5x − 3 = 0, b is −5, not 5.

The part under the root, b² − 4ac, is the discriminant, and it predicts the answer before you finish:

• positive → two real solutions
• zero → one repeated solution
• negative → no real solutions

Substitute carefully with parentheses. Most errors here are sign errors, not formula errors.

Derived by completing the square on ax² + bx + c = 0, the formula gives both roots for any a ≠ 0 over ℝ or ℂ.

The discriminant Δ = b² − 4ac determines the nature of the roots: Δ > 0 gives two distinct real roots (and if Δ is a perfect square with rational coefficients, they are rational and the quadratic factors over ℚ); Δ = 0 gives one real root of multiplicity 2, where the parabola is tangent to the x-axis; Δ < 0 gives a conjugate pair of complex roots p ± qi.

Vieta's formulas follow directly from the two roots: their sum is −b/a and their product is c/a. These give a quick check on any computed pair and are the fastest way to reconstruct a quadratic from its roots.

Numerically, the formula suffers catastrophic cancellation when b² ≫ 4ac and the signs align, which is why computational implementations use the algebraically equivalent form 2c/(−b ∓ √Δ) for one of the roots.