Complex Equation Input
Choose the equation type, enter coefficients, then solve for complex roots or the linear complex solution.
Plotly Graph
This Argand plane marks root positions and, for quadratic mode, the discriminant point.
Example Data Table
| Equation Type | Equation | Expected Output | Note |
|---|---|---|---|
| Linear | (2 + i)z + (4 - 6i) = 0 | z = -0.4 + 3.2i | Useful for validating the linear branch. |
| Quadratic | z² + 2z + 5 = 0 | z = -1 ± 2i | Real coefficients still produce complex roots. |
| Quadratic | z² - 2iz + 1 = 0 | z = (1 ± √2)i | Shows purely imaginary solutions. |
Formula Used
Linear complex equation: bz + c = 0
Solution: z = -c / b
Quadratic complex equation: az² + bz + c = 0
Solutions: z = (-b ± √(b² - 4ac)) / 2a
Supporting measures:
Modulus: |z| = √(x² + y²)
Argument: arg(z) = atan2(y, x)
Polar form: z = r∠θ
For a complex square root, the solver uses the principal square root, then applies the plus and minus branches in the quadratic formula.
How to Use This Solver
- Select linear or quadratic mode.
- Enter each coefficient's real and imaginary parts.
- Click Solve Equation.
- Read the result table placed above the form.
- Review modulus, arguments, and verification checks.
- Inspect the Argand plane graph for root placement.
- Use CSV or PDF buttons to export the result summary.
Frequently Asked Questions
1. What equations does this solver handle?
It solves linear equations of the form bz + c = 0 and quadratic equations of the form az² + bz + c = 0, where every coefficient may contain real and imaginary parts.
2. Can the coefficients be complex numbers?
Yes. Each coefficient accepts separate real and imaginary values. That allows inputs such as 3 - 2i, -4 + i, or purely imaginary values like 5i.
3. Why do I see two roots for a quadratic?
Quadratic equations normally produce two solutions because the square root step has positive and negative branches. Repeated roots can still appear when the discriminant becomes zero.
4. What does the discriminant mean here?
The discriminant is b² - 4ac. In complex algebra, it still controls the square root part of the quadratic formula and influences whether both roots match or separate.
5. What is modulus and argument?
Modulus is the distance from the origin on the complex plane. Argument is the angle from the positive real axis. Together, they describe the number in polar form.
6. Why does the graph use an Argand plane?
An Argand plane displays the real part horizontally and the imaginary part vertically. That makes root locations easy to interpret, compare, and verify visually.
7. What happens if a is zero?
If quadratic mode is selected and a equals zero, the equation is no longer quadratic. The solver shows a validation message and asks for linear mode or a nonzero a value.
8. Can I export the result summary?
Yes. After solving, use the CSV button for spreadsheet-ready text or the PDF button for a clean report that includes the computed result table.