Analyze coefficients, roots, and discriminants with clarity. Review exact forms, decimals, and graph-ready output instantly. Built for students needing reliable quadratic solutions every time.
Use values like 2, -3.5, 4i, or 3-2i.
| Equation | Coefficients | Discriminant | Expected roots |
|---|---|---|---|
| x² + 2x + 5 = 0 | a = 1, b = 2, c = 5 | -16 | -1 ± 2i |
| x² + (2+i)x + (1-i) = 0 | a = 1, b = 2+i, c = 1-i | -1 + 8i | Complex pair from formula |
| (1-i)x² + 4x + 8 = 0 | a = 1-i, b = 4, c = 8 | -16 - 32i | Distinct complex roots |
When coefficients are complex, the same quadratic formula still applies. The only extra step is evaluating the square root of a complex discriminant.
This calculator uses the principal complex square root, then computes both roots, the root sum, and the root product for consistency checks.
5i or 3-2i.Complex quadratic equations appear in algebra, physics, control systems, and signal analysis. Their coefficients may contain both real and imaginary parts, so input quality matters from the first step. A practical solver must accept values such as 4, -3i, or 2-5i clearly. Clear formatting reduces entry errors and supports faster checking when users compare equations in one session.
The discriminant remains the most informative intermediate result in quadratic solving. For complex equations, Δ = b² − 4ac still determines how the two roots separate after the square root is evaluated. Its real component influences horizontal movement on the complex plane, while its imaginary component affects angular placement. Displaying both Δ and √Δ helps users understand why roots move, merge, or rotate when coefficient values change.
Each root can be interpreted as a coordinate pair made of a real part and an imaginary part. This geometric view is useful because symbolic answers become visible points. A positive imaginary value places a root above the horizontal axis, and a negative value places it below. When both roots are plotted together, users can quickly inspect spacing, symmetry, and the midpoint produced by the equation structure.
Complex arithmetic often produces long decimal expansions after multiplication, division, and square-root operations. Precision control allows the same solution to serve different needs. Lower precision improves readability in homework or presentations, while higher precision supports verification and technical review. A strong solver should calculate numerically first and apply rounding only at display time. That approach preserves internal consistency across roots, discriminants, and validation fields.
Advanced calculators should help users verify results, not only produce them. The sum and product of the roots provide a convenient Vieta check. If coefficients are correct, these values should align with theory. When they do not, users can recheck signs, imaginary terms, or decimal placement. This makes the solver especially useful for study, testing, and auditing.
This calculator supports classroom practice and applied problem solving together. Students can test examples, compare manual steps, and strengthen intuition about complex roots. Analysts can evaluate characteristic equations or parameter changes without rebuilding calculations repeatedly. The graph, export tools, formula panel, and example table create a compact reference environment. That combination improves clarity and confidence during daily technical work.
The solver accepts real or complex coefficients written as values like 2, -3.4, 5i, or 3-2i. The leading coefficient must not be zero.
Yes. The same formula applies. The difference is that the discriminant and its square root are evaluated using complex arithmetic instead of real-only arithmetic.
Showing the discriminant helps users understand how the coefficients shape the roots. It also makes checking intermediate steps easier during classroom work or technical review.
The graph plots both roots and the midpoint on the complex plane. Horizontal position shows the real part, and vertical position shows the imaginary part.
Yes. You can export result values to CSV for spreadsheet work or use the print-based PDF option for saving a clean record.
They provide a quick validation check. For a correct solution, these values should match the relationships predicted by Vieta’s formulas for the entered coefficients.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.