What This Quadratic Solver Does
Enter the three coefficients of a quadratic equation in standard form —
— and the solver returns the roots (real or complex), the discriminant with its interpretation, and the vertex of the parabola, showing each substitution step so you can follow or check homework line by line.
The Quadratic Formula
It solves every quadratic — derived by completing the square on the general form. The action is inside the square root, called the discriminant:
| Δ | Roots | Graphically |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One repeated real root | Vertex touches the axis |
| Δ < 0 | Two complex conjugate roots | No axis crossing |
The solver computes Δ first, tells you which case you're in, then produces the roots accordingly — including complex ones in a ± bi form.
Worked Examples, One Per Case
Δ > 0 — x² − 5x + 6 = 0
Check by factoring: (x−2)(x−3) = 0 ✓
Δ = 0 — x² − 6x + 9 = 0
The perfect square (x−3)² = 0 — the parabola kisses the axis at its vertex.
Δ < 0 — x² + 2x + 5 = 0
Complex conjugates — always a mirrored pair when coefficients are real.
Vertex, Axis of Symmetry, and Reading the Parabola
Axis of symmetry: the vertical line x = −b/2a
For y = x² − 5x + 6: vertex at (2.5, −0.25) — the minimum, since a > 0 opens upward (a < 0 would open downward with a maximum). The roots, when real, sit symmetrically around the axis — a useful sanity check on any solution.
Two more identities worth knowing (Vieta's formulas): sum of roots = −b/a, product = c/a. For x² − 5x + 6: sum 5, product 6 → roots 2 and 3 confirmed without re-solving.
Where Quadratics Show Up
- Projectile motion: height h = −½gt² + v₀t + h₀ — "when does it land?" is a quadratic root.
- Optimization: maximum revenue/area problems peak at the vertex.
- Geometry: the Triangle Calculator's law-of-cosines cases reduce to quadratics.
- Finance: break-even and rate problems (see the Finance Calculator) often quadratic in disguise.
Solving methods ranked by convenience: factoring (when integers cooperate) → formula (always works) → completing the square (best for deriving vertex form).
The Bottom Line
One formula, one discriminant, and every quadratic in existence surrenders: Δ tells you what kind of roots to expect, the formula produces them, and the vertex completes the graph's story. Use the solver above for the arithmetic — and keep the discriminant table in your head for the ten-second version of any quadratic question.