Quadratic Formula Calculator — quick guide
1) Core idea
Solve ax² + bx + c = 0 using x = (−b ± √(b² − 4ac)) / (2a) with a ≠ 0. The discriminant Δ = b² − 4ac determines the nature of roots.
2) How this tool works
- Inputs: coefficients a, b, c.
- Computes Δ and returns two real roots, a repeated root (Δ=0), or two complex conjugates (Δ<0).
- Also shows step-by-step work and parabola properties (vertex, axis, intercepts).
3) Sanity checks
- If a = 0 → not quadratic; tool rejects it.
- Δ > 0 → two distinct real roots; Δ = 0 → one repeated real root; Δ < 0 → complex roots.
- Symmetry: swapping both roots preserves sum = −b/a and product = c/a (Vieta).
4) Shortcuts that help
- Try factoring when coefficients are small integers.
- Complete the square to reveal the vertex quickly.
- Normalize by dividing all terms by a (doesn’t change roots).
5) Common pitfalls
- Forgetting the “−b” sign or miscomputing 4ac.
- Rounding too early; this tool rounds only for display.
- Assuming complex roots are “errors”—they’re valid when Δ < 0.
6) Micro-examples
- x² − 5x + 6 = 0 → roots 2 and 3 (Δ = 1).
- x² − 2x + 1 = 0 → root 1 (double root, Δ = 0).
- x² + 1 = 0 → roots ±i (Δ = −4).
7) Mini-FAQ
- Complex output? Shown as a ± bi with standard double precision.
- Exact vs decimal? Results are numeric; keep radicals by doing algebra manually if needed.
- Precision? Internally full precision; display rounds.
8) Action tip
Check Δ first to anticipate root types, then verify each root by substitution into ax²+bx+c.