Quick start
Choose Quadratic for ax² + bx + c or Polynomial for degree up to 3.
Enter coefficients and click Calculate.
Read roots first, then use the graph for a visual check.
Static example: for x² - 3x + 2 = 0, the discriminant is 1, so the roots are x=1 and x=2. The graph should cross the x-axis at those two points.
Results
- Roots
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- Discriminant
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- Classification
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- Factorization
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- Completing the square
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- Vertex & axis
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How it's calculated
Graph
Turn this on to show short reminders about discriminants, root checks, and rounding.
FAQ
What steps does the quadratic mode display?
It starts with the discriminant. Then it shows formula substitution, optional root simplification, vertex form, and factorization.
How does polynomial mode handle degree 3?
For degree-3 polynomials, it checks rational root candidates first. Next it runs synthetic division and solves the remaining quadratic.
Which solver mode should I use first?
Use quadratic mode for ax^2+bx+c, polynomial mode for higher-degree expressions, and factoring mode when you want to inspect rational roots or factor form before decimal approximations.
Why can another solver list different-looking roots?
Solvers may order roots differently, keep radicals exact, switch to decimals, or factor over different number sets. Compare the roots by substituting them back into the original equation.
Which result should I use when exact and decimal answers differ?
Exact forms and step logs are the preferred result when available. Decimal roots are rounded for display, so use exact radicals or factor form when precision matters.
Quadratic and polynomial solving notes
Start with standard form
Put the equation in one variable and move all terms to one side. This makes coefficients, degree, discriminant, and factor checks easier to verify.
Quadratic steps
For degree 2, review the discriminant first. It explains whether the roots are real, repeated, or complex before the formula is applied.
Polynomial mode
For higher-degree polynomials, rational-root checks and synthetic division can reduce the problem before numeric methods are needed.
Common mistakes
Do not drop a leading zero coefficient, change signs while moving terms, or compare rounded decimal roots without substitution.
Checking answers
Substitute each root into the original equation. For factored forms, expand the factors to confirm they reproduce the polynomial.