A walkthrough, end to end.
- 1
Enter the three coefficients a, b, and c.
- 2
The calculator computes the discriminant b² − 4ac and applies the quadratic formula.
- 3
Result includes both roots (real or complex) plus the discriminant interpretation.
Quadratic formula
x = (−b ± √(b² − 4ac)) / 2a. Discriminant b² − 4ac determines the nature: positive = two real roots, zero = one repeated real root, negative = two complex conjugate roots.
What you can do with this.
Solve any quadratic equation
x² + 5x + 6 = 0 → x = −2 or x = −3. Standard high-school algebra workhorse. Works for any a, b, c with a ≠ 0.
Discriminant analysis
b² − 4ac > 0: two distinct real roots. = 0: one repeated real root (parabola tangent to x-axis). < 0: complex conjugate roots. The calculator reports which case.
Projectile motion
Height equation h = −16t² + v₀t + h₀ (US units). Solving h = 0 gives time-of-flight. The calculator's quadratic solve is the math behind every artillery tutorial.
Profit / break-even analysis
Profit = revenue − cost as quadratic in price. Solving = 0 finds break-even prices; vertex of parabola is the profit-maximizing price.
Geometry: area constraints
If perimeter is fixed and area is given, side lengths satisfy a quadratic. Useful for landscaping, packaging optimization, and AP Calc problems.
Complex roots interpretation
When discriminant < 0, no real solutions exist — but complex roots indicate the parabola never crosses the x-axis. Common in physics oscillation problems.
Factoring vs. quadratic formula
When roots are integers, factoring is faster (e.g., x² + 5x + 6 = (x+2)(x+3)). For non-integer or irrational roots, quadratic formula is the only direct approach.
Quadratic formula 2026 — what's current
Standard tool from middle school onward. AI assistants (ChatGPT, Claude) compute reliably. Standalone calculator wins for speed when you don't want to wait for a chat response.
Frequently asked.
Then it's not a quadratic — it's linear. The formula doesn't apply (would divide by zero). The calculator returns an error in this case.
When discriminant < 0, you take √ of a negative number. Result has form p ± qi where i = √−1. The calculator displays both real and imaginary parts.
y = a(x−h)² + k where (h, k) is the vertex. Different from standard form ax² + bx + c. To convert: h = −b/(2a), k = c − b²/(4a).
No. Calculations run entirely in your browser.