Calculation details
Planning estimate only, not financial, tax, or legal advice. Verify assumptions and current rules before making decisions.
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What does this calculator estimate?
A quadratic formula calculator solves ax² + bx + c = 0 for x. Enter the coefficients a, b, and c to get the roots — real or complex — plus the discriminant.
- Roots = (−b ± √(b² − 4ac)) ÷ 2a
- Discriminant > 0: two real roots; = 0: one; < 0: complex
- Sum of roots = −b/a, product = c/a
What the quadratic formula does
The quadratic formula finds the x-values where a parabola crosses the x-axis. The discriminant tells you how many crossings exist: two real roots when it's positive, one (touching) when zero, and none (complex pair) when negative.
Limitations to watch for
The tool requires a ≠ 0 — otherwise the equation is linear, not quadratic. Complex roots appear in the form 'real ± imaginary·i'. Rounding can hide near-double roots; the discriminant near zero is sensitive to input precision.
How to use it in practice
Use it to solve physics, finance, and geometry problems that reduce to ax² + bx + c = 0 — projectile motion, break-even, and area problems. Check the discriminant first to know whether real solutions exist. Verify by plugging roots back into the equation.
['Enter the coefficients a, b, and c.', 'The tool computes the discriminant and the roots.', 'Complex roots are shown as real ± imaginary.']
How this calculator works
Formula
For ax² + bx + c = 0, roots = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant D = b² − 4ac decides the roots: D > 0 gives two real roots, D = 0 a double root, D < 0 complex roots. Sum of roots = −b/a, product = c/a.
Worked example
x² − 5x + 6 = 0: discriminant = 25 − 24 = 1, so roots = (5 ± 1) ÷ 2 = 3 and 2.
Assumptions to verify
- The equation is in standard form ax² + bx + c = 0.
- a is non-zero.
- Coefficients are real numbers.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a — the general solution to ax² + bx + c = 0.
What is the discriminant?
D = b² − 4ac. D > 0 means two real roots, D = 0 one double root, D < 0 a pair of complex roots.
What if a is zero?
The equation is linear (bx + c = 0), not quadratic — the formula doesn't apply, and this tool rejects it.
What are complex roots?
When the discriminant is negative, the square root is imaginary and the roots are a complex pair (real ± imaginary·i). They mean the parabola never crosses the x-axis.
How do I check my answer?
Plug each root back into ax² + bx + c — it should equal zero. The sum of roots equals −b/a and the product equals c/a as a quick cross-check.
What does the vertex have to do with it?
The roots are symmetric around the vertex x = −b/(2a). The discriminant tells you whether the vertex sits below, on, or above the x-axis.
Can the formula solve any quadratic?
Yes, for any real coefficients with a ≠ 0 — real or complex solutions included.
Cite this tool
BoringToolsKit. “Quadratic Formula Calculator.” boringtoolskit.com/quadratic-formula-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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