Math
Quadratic Solver
Solve ax² + bx + c = 0 in seconds, including discriminant analysis and whether roots are real or complex.
Solve ax² + bx + c = 0 with discriminant analysis.
Quadratic formula
x = [−b ± √(b² − 4ac)] ÷ (2a)
The discriminant Δ = b² − 4ac reveals the nature of roots: Δ > 0 gives two real roots, Δ = 0 gives a repeated real root, and Δ < 0 yields complex conjugates.
How to use
- Enter coefficients a, b, and c for your quadratic equation.
- Ensure a ≠ 0; otherwise the equation is linear and cannot use the quadratic formula.
- Read the discriminant, roots, and root nature in the results panel.
Example
Input: a = 1, b = -3, c = 2
Output: Discriminant = 1, Roots = 2 and 1 (two real roots)
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for Quadratic Solver: quadratic equation solver, quadratic formula calculator, quadratic solver, quadratic equation calculator. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
Quadratic Solver: Roots, discriminant, and root nature for quadratic equations. It's built around algebra, roots, quadratic, so you can go from a raw question to a checked answer without switching tools.
The math behind it: The discriminant Δ = b² − 4ac reveals the nature of roots: Δ > 0 gives two real roots, Δ = 0 gives a repeated real root, and Δ < 0 yields complex conjugates. The core relationship is x = [−b ± √(b² − 4ac)] ÷ (2a), shown above the calculator so you can see exactly how your inputs turn into the result.
To use it well: (1) Enter coefficients a, b, and c for your quadratic equation. (2) Ensure a ≠ 0; otherwise the equation is linear and cannot use the quadratic formula. (3) Read the discriminant, roots, and root nature in the results panel. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering a = 1, b = -3, c = 2 returns Discriminant = 1, Roots = 2 and 1 (two real roots). Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
What happens if a = 0?
The calculator treats the expression as invalid because the quadratic formula requires a non-zero a coefficient. Adjust your equation or switch to a linear solver.
How are complex roots displayed?
When the discriminant is negative, roots are shown as a conjugate pair in the form real ± imaginary·i.
What formula does the Quadratic Solver use?
The discriminant Δ = b² − 4ac reveals the nature of roots: Δ > 0 gives two real roots, Δ = 0 gives a repeated real root, and Δ < 0 yields complex conjugates.
How do I use the Quadratic Solver?
Enter coefficients a, b, and c for your quadratic equation. Ensure a ≠ 0; otherwise the equation is linear and cannot use the quadratic formula. Read the discriminant, roots, and root nature in the results panel.