Enter the coefficients of a quadratic (ax² + bx + c = 0) or cubic (ax³ + bx² + cx + d = 0) equation to find its real and complex roots, using the quadratic formula or Cardano's method.
Results
Calculated
Root 1 (x₁)
—
First root of the equation
Root 2 (x₂)
—
Second root of the equation
Root 3 (x₃)
—
Third root — cubic equations only
Root classification
—
How many roots are real vs. complex
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How to use this calculator
Choose Quadratic or Cubic from the degree menu, enter the coefficients a, b, c (and d for a cubic), then click Calculate. The calculator solves ax² + bx + c = 0 with the quadratic formula, or ax³ + bx² + cx + d = 0 with Cardano's method, and reports every root — real or complex. Click Clear to reset the fields to the example equation.
The quadratic formula
For ax² + bx + c = 0 (a ≠ 0), the roots are x = (−b ± √(b² − 4ac)) / (2a). The discriminant Δ = b² − 4ac tells you what kind of roots to expect: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives two complex conjugate roots.
Cardano's method for cubics
For ax³ + bx² + cx + d = 0 (a ≠ 0), the equation is first divided by a and depressed (the x² term removed) with the substitution x = t − b/(3a), giving t³ + pt + q = 0. Cardano's formula and, when three real roots exist, a trigonometric form of it (to avoid complex intermediate values) then solve for t, and the roots are shifted back to x. Every cubic with real coefficients has at least one real root; it may have one real root plus a complex conjugate pair, or three real roots (possibly repeated).
Interpreting the results
Root 1 and Root 2 are always populated. Root 3 only applies to cubic equations — it shows a dash for quadratics, which have exactly two roots. Complex roots are shown in the form a ± bi, where i is the imaginary unit (i² = −1). Root classification summarizes how many roots are real versus complex, based on the sign of the discriminant.
Frequently Asked Questions
What is the quadratic formula?
For ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a). The term under the square root, b² − 4ac, is called the discriminant: positive means two distinct real roots, zero means one repeated real root, and negative means two complex conjugate roots.
How does the calculator solve a cubic equation?
It uses Cardano's method: the cubic ax³ + bx² + cx + d = 0 is divided by a and shifted with x = t − b/(3a) to remove the squared term, producing a depressed cubic t³ + pt + q = 0. That simpler form is solved with Cardano's radical formula, or a trigonometric equivalent when there are three real roots, and the results are shifted back to give x.
What does a complex root mean?
A complex root has the form a + bi, where i = √(−1). Complex roots of a real-coefficient polynomial always come in conjugate pairs, a + bi and a − bi. They are valid mathematical solutions, but they do not correspond to a point where the graph of the polynomial crosses the x-axis — only real roots do that.
Does every cubic equation have a real root?
Yes. Because a cubic's graph goes from −∞ to +∞ (or the reverse) as x moves across all real numbers, it must cross the x-axis at least once. So every cubic with real coefficients has at least one real root, with the other two either both real or a complex conjugate pair.
Practical Guide for Polynomial Roots Calculator - Find Zeros of Quadratic & Cubic Equations
Polynomial Roots Calculator - Find Zeros of Quadratic & Cubic Equations is most useful when the inputs reflect the situation you are actually planning around, not a best-case estimate. Treat the result as a decision aid: it gives you a structured way to compare assumptions, spot outliers, and decide what to verify next. For Math work, the most important review lens is formula choice, units, rounding, weighting, and the exact meaning of each input.
Start with a baseline run using values you can defend. Then change one assumption at a time and watch which output moves the most. If one input dominates the result, spend your verification time there first. If several inputs have similar influence, use a conservative scenario and an optimistic scenario to create a practical range instead of relying on a single exact number.
Before acting on the result, verify the result with a manual calculation or a second method when the output affects grades, budgets, or engineering work. This is especially important when the calculator supports a purchase, project plan, performance target, or operational decision. The calculator can make the math consistent, but the quality of the conclusion still depends on current data, clear units, and assumptions that match your real constraints.
When the output looks surprising, slow down and inspect each input in order. A small change in one high-leverage field can move the final number more than several low-leverage fields combined. For Polynomial Roots Calculator - Find Zeros of Quadratic & Cubic Equations, that means you should first confirm the value with the greatest scale, then confirm the value with the greatest uncertainty, then rerun the calculator with conservative and optimistic assumptions. This sequence turns the calculator from a single answer into a practical decision range.
Review Checklist
Confirm every input uses the unit and time period requested by the calculator.
Run a low, expected, and high scenario so the answer has a useful range.
Check whether rounding or a missing decimal place changes the decision.
Update the calculation after each new value is known or whenever the formula structure changes.