How this calculator works
Every quadratic equation ax² + bx + c = 0 with a ≠ 0 can be solved with the quadratic formula. The discriminant b² - 4ac decides the kind of answer: positive gives two different real roots, zero gives one repeated root, and negative gives a complex conjugate pair. Roots are shown in ascending order for real solutions.
Formula
Δ = b² - 4ac and x = (-b ± √Δ) / (2a). When Δ < 0, x = -b/(2a) ± (√(-Δ) / (2|a|))i.
Worked example
For x² - 5x + 6 = 0, Δ = 25 - 24 = 1, so x = (5 ± 1)/2, which gives the roots 2 and 3. For x² - 4x + 4 = 0, Δ = 0 and the double root is 2. For x² + 2x + 5 = 0, Δ = -16 and the roots are -1 ± 2i.
Advertisement
Frequently asked questions
Why must a not be zero?
The formula divides by 2a. With a = 0 the equation collapses to bx + c = 0, a linear equation with at most one root, so it is not solved here.
What is a double root?
When the discriminant is exactly zero, both branches of the formula give the same value. The parabola touches the x axis at that single point.
How accurate are the roots?
Roots are shown with up to 6 significant digits and trailing zeros dropped. Floating-point arithmetic has finite precision, so very large or very small coefficients may not give reliable roots.