How the quadratic formula is used
The calculator solves equations in the form ax^2 + bx + c = 0. It first calculates the discriminant, D = b^2 - 4ac. If D is positive, there are two real roots. If D is zero, there is one repeated real root. If D is negative, the roots are complex.
Why this is more useful than only showing roots
Many simple solvers return only x-values. This page also shows the discriminant and vertex, which helps users check whether the parabola opens upward or downward, where it turns, and whether real x-intercepts should exist.
Example
For x^2 - 3x + 2 = 0, the discriminant is 1, so the equation has two real roots: x = 1 and x = 2. The vertex is at x = 1.5.
Limitations
- The calculator solves single-variable quadratic equations only.
- Very large coefficients can lose precision because browser JavaScript uses floating-point arithmetic.
- If a = 0, use a linear equation calculator instead.
References
- NIST/SEMATECH e-Handbook: Quadratic Polynomial - polynomial model context, accessed 2026-05-17.
FAQ
What does the discriminant tell me?
The discriminant b squared minus 4ac tells whether a quadratic has two real roots, one repeated real root, or two complex roots.
Can coefficient a be zero?
No. If a is zero, the equation is linear rather than quadratic.
Why show the vertex too?
The roots show where the parabola crosses the x-axis, while the vertex shows the turning point and helps users understand the graph shape.
Last reviewed: 2026-05-17