Solve quadratic equations in the three most common forms: standard form Ax² + Bx + C = 0, vertex form A(x − H)² + K = 0, or factored form A(x − x₁)(x − x₂) = 0. The calculator shows the discriminant, the roots, the parabola graph, and the step-by-step solution.
Quadratic Equation Calculator
Select the formula and enter the parameters
How to use the calculator
- Standard form: Enter A, B and C from Ax² + Bx + C = 0. The calculator uses the quadratic formula to find the roots.
- Vertex form: If you know the vertex (H, K) and the leading coefficient A, enter them directly to get the standard form and the roots.
- Factored form: If you know the roots x₁ and x₂, enter them to get the coefficients and the graph.
- Check Allow negative discriminant to display complex roots when Δ < 0.
Formulas
| What to calculate | Formula |
|---|---|
| Discriminant | Δ = B² − 4AC |
| Roots (quadratic formula) | x = (−B ± √Δ) ÷ (2A) |
| Vertex x-coordinate | H = −B ÷ (2A) |
| Vertex y-coordinate | K = C − B² ÷ (4A) |
| Vertex form | f(x) = A(x − H)² + K |
Frequently asked questions
What is the quadratic formula?
The quadratic formula finds the roots of Ax² + Bx + C = 0: x = (−B ± √Δ) ÷ (2A), where Δ = B² − 4AC. The name "Bhaskara's formula" (used in Portuguese-speaking countries) honours Indian mathematician Bhaskara II (12th century), though the complete formula was developed by various mathematicians throughout history.
What does the discriminant Δ mean?
Δ = B² − 4AC determines the nature of the roots:
- Δ > 0: two distinct real roots
- Δ = 0: one double real root (the parabola touches the x-axis at a single point)
- Δ < 0: no real roots (the parabola does not intersect the x-axis)
How do you convert to vertex form?
Complete the square: A(x² + (B/A)x) + C = A(x + B/(2A))² − B²/(4A) + C. The vertex is at H = −B/(2A), K = C − B²/(4A). The vertex form is f(x) = A(x − H)² + K.
When does the parabola open upward or downward?
The leading coefficient A determines the concavity: A > 0 → opens upward (∪), the vertex is the minimum point. A < 0 → opens downward (∩), the vertex is the maximum point.
See also…
