Cubic Equation Solver

Solve ax³ + bx² + cx + d = 0 step by step — free cubic equation solver with worked examples, real-world defaults and instant answers.

ax³ + bx² + cx + d = 0

a (x³)b (x²)c (x)d

Real root x11

Real root x22

Real root x33

1
Substitute x = t − b/3a to remove the x² term (depressed cubic):
t³ + pt + q = 0, p = -1, q = 0, shift = 2
2
Compute the cubic discriminant Δ = −(4p³ + 27q²):
Δ = 4 → three distinct real roots
3
Use the trigonometric method: tₖ = 2√(−p/3)·cos(⅓·acos(3q/(p·2√(−p/3))) − 2πk/3):
x1 = 1, x2 = 2, x3 = 3
4
Verify by substitution into the original cubic:
f(1) = 0 ≈ 0 ✓ f(2) = 0 ≈ 0 ✓ f(3) = 0 ≈ 0 ✓

Every cubic with real coefficients has at least one real root — the graph of an odd-degree polynomial must cross the x-axis. The default equation x³ − 6x² + 11x − 6 factors as (x−1)(x−2)(x−3), a classic worked example.

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Cubic equation solver — real roots of ax³ + bx² + cx + d = 0 (trigonometric / Cardano method).

About Cubic Equation Solver

The Cubic Equation Solver solves ax³ + bx² + cx + d = 0 for you with the full working shown — every substitution, every unit, every step, exactly the way a good teacher writes it on the board. Every cubic with real coefficients has at least one real root — the graph of an odd-degree polynomial must cross the x-axis.

How to use Cubic Equation Solver

1Enter your known values in the input fields (sensible real-world defaults are pre-filled).
2The tool substitutes them into ax³ + bx² + cx + d = 0 and recomputes live.
3Read the answer in the result box, then expand the step-by-step solution to see the full working.
4Copy the method into your notebook, or change inputs to explore how the result behaves.

Why use Cubic Equation Solver?

✓Solves ax³ + bx² + cx + d = 0 instantly as you type — no submit button, no waiting
✓Step-by-step solution shown for every calculation, not just the final answer
✓Realistic example values pre-loaded so you can see a worked example immediately
✓100% free, no sign-up, runs entirely in your browser — your numbers never leave your device
✓Mobile-friendly and fast enough to use mid-homework or mid-lesson

Frequently asked questions

What formula does the Cubic Equation Solver use?+

It uses ax³ + bx² + cx + d = 0. The steps section shows the formula with your actual numbers substituted, so you can follow (and verify) every stage of the calculation rather than trusting a black box.

What should I keep in mind when using this calculator?+

Is the Cubic Equation Solver free to use?+

Completely free, with no sign-up or limits. All computation happens client-side in your browser, so it is fast, private, and works even on slow connections once the page has loaded.

Can I use this for homework, teaching or exam prep?+

That is exactly what it is built for. The step-by-step working mirrors how solutions are presented in class, so you can check homework, build worked examples for teaching, or practise method recall before exams.

Embed Cubic Equation Solver on your website

Want Cubic Equation Solveron your own site? Paste this snippet into any HTML page — it's free, with no API key or sign-up. The tool loads in an iframe and keeps working exactly as it does here.

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