Enter System Coefficients
Example Data Table
| Parameter | Example value | Meaning |
|---|---|---|
| a₁ | 1 | Circle x² + y² − 25 = 0 |
| b₁ | 0 | No mixed xy term in equation 1 |
| c₁ | 1 | Quadratic weight on y² |
| d₁, e₁, f₁ | 0, 0, −25 | Circle radius equals 5 |
| a₂, b₂, c₂ | 1, 0, 0 | Parabolic curvature on x² only |
| d₂, e₂, f₂ | −2, −1, −4 | Second equation x² − 2x − y − 4 = 0 |
| Search range | −6 to 6 | Grid of starting points for Newton updates |
| Expected intersections | Up to 4 | Two quadratic curves can meet at four real points |
Formula Used
Each equation uses the general quadratic form:
F₁(x, y) = a₁x² + b₁xy + c₁y² + d₁x + e₁y + f₁
F₂(x, y) = a₂x² + b₂xy + c₂y² + d₂x + e₂y + f₂
The solver seeks points where both expressions equal zero simultaneously. It applies Newton’s method to the vector system:
J(x, y) · Δ = −F(x, y)
where J is the Jacobian matrix:
∂F₁/∂x = 2a₁x + b₁y + d₁, ∂F₁/∂y = b₁x + 2c₁y + e₁
∂F₂/∂x = 2a₂x + b₂y + d₂, ∂F₂/∂y = b₂x + 2c₂y + e₂
The residual norm is √(F₁² + F₂²). Smaller residuals mean the reported intersection satisfies both equations more accurately.
How to Use This Calculator
- Enter six coefficients for the first quadratic equation.
- Enter six coefficients for the second quadratic equation.
- Choose a search window that likely contains intersections.
- Set seed density higher when curves are difficult or tangent.
- Use a smaller tolerance for stricter accuracy checks.
- Click Solve Quadratic System to calculate intersections.
- Review solutions, residuals, starting seeds, and Jacobian values.
- Export the results table as CSV or PDF when needed.
FAQs
1. What does this calculator solve?
It solves two quadratic equations in two variables. The tool searches for real intersection points that satisfy both equations together.
2. Why can there be several answers?
Two quadratic curves may intersect at zero, one, two, three, or four real points. Geometry and coefficient values determine the final count.
3. What is seed density?
Seed density controls how many starting guesses are tested across the search window. More seeds improve discovery, especially for separated or delicate intersections.
4. What does residual norm mean?
Residual norm measures how close both equations are to zero at a reported point. Smaller values indicate a more accurate numerical intersection.
5. Why was no solution found?
Your curves may not intersect in real coordinates, or intersections may lie outside the selected search range. Increase the range or seed density.
6. What does the Jacobian determinant show?
It indicates local sensitivity near a solution. Very small determinants suggest tangent behavior, instability, or cases where Newton updates become unreliable.
7. Can this tool find complex solutions?
No. This version reports real intersections only. Complex-valued solutions require symbolic algebra or a solver designed for complex arithmetic.
8. When should I use tighter tolerance?
Use tighter tolerance when you need cleaner verification, research-grade checking, or when small coefficient changes noticeably affect the intersection coordinates.