The quadratic formula
Every equation of the form ax² + bx + c = 0 (with a ≠ 0) is solved by
x = (−b ± √(b² − 4ac)) ÷ 2a
The part under the root, Δ = b² − 4ac, is the discriminant. It tells you what kind of answer to expect: Δ > 0 gives two real roots, Δ = 0 one repeated root, and Δ < 0 two complex roots, because the parabola never touches the x-axis.
Worked example
x² − 3x + 2 = 0: a = 1, b = −3, c = 2. Δ = 9 − 8 = 1, so x = (3 ± 1) ÷ 2, giving x₁ = 1 and x₂ = 2. Check: (x − 1)(x − 2) = x² − 3x + 2. The vertex is at x = 1.5, where y = −0.25.
With a = 1, b = 2, c = 5: Δ = 4 − 20 = −16, so x = −1 ± 2i.
How this works
The page computes the discriminant and, when it is positive, uses an algebraically equivalent version of the quadratic formula that avoids subtracting two nearly equal numbers, so tiny roots stay accurate even when b is huge. A zero discriminant gives one double root; a negative one gives a pair of complex conjugate roots written as p ± qi. It also reports the vertex of the parabola, (−b/2a, f(−b/2a)), and the axis of symmetry. Everything runs in your browser.
Frequently asked
What does the discriminant tell me?
Positive: two real roots. Zero: one double root. Negative: two complex roots.
What if a is 0?
Then the equation is linear (bx + c = 0) and x = −c/b.
What is the vertex of a parabola?
Its turning point: (−b/2a, f(−b/2a)). It is a minimum if a > 0 and a maximum if a < 0.
Can it factor the equation?
When the roots are r₁ and r₂, the factored form is a(x − r₁)(x − r₂).
Sources
Reviewed Oct 17, 2026 · test vectors fixtures/calculators/math/quadratic-equation-solver/