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Cubic Equation Solver

How to Solve Cubic Equations

A practical guide to solving cubic equations, from quick factoring checks to the full Cardano method.

Table of Contents

1

Factoring by Inspection

Start by looking for common factors, grouping opportunities, or known identities.

  • Common factor: x³ - 4x = x(x² - 4) = x(x - 2)(x + 2)
  • Grouping: x³ + x² - x - 1 = x²(x+1) - (x+1) = (x+1)(x² - 1)
  • Difference of cubes: x³ - a³ = (x - a)(x² + ax + a²)
2

Rational Root Theorem

If a polynomial with integer coefficients has a rational root p/q, then p divides the constant term and q divides the leading coefficient.

p divides d and q divides a

For example, for 2x³ - 3x² - 8x + 12 = 0, the possible rational roots come from the factors of 12 over the factors of 2.

3

Synthetic Division

Once a rational root r is found, divide the cubic by (x - r) to reduce the problem to a quadratic.

ax³ + bx² + cx + d = (x - r)(ax² + b₁x + c₁)

4

Cardano's Method

  1. Normalize the cubic.
  2. Substitute x = t - b/(3a) to remove the quadratic term.
  3. Solve the depressed cubic t³ + pt + q = 0.
  4. Compute Delta = q²/4 + p³/27.
  5. Apply the matching root formula and convert back to x.

Our cubic equation solver automates this process and shows each step in a cleaner format.

5

Trigonometric Method

When Delta is negative, the cubic has three distinct real roots and the trigonometric form is often the clearest route.

Set r = 2 sqrt(-p/3) and theta = (1/3) arccos(-q / (2 sqrt(-p³/27))), then build the three roots from cosine shifts.

6

Numerical Approximation

When an exact symbolic form is not required, numerical methods provide quick approximations.

  • Newton-Raphson: xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)
  • Bisection: shrink an interval where the function changes sign
  • Graphical estimation: inspect where the cubic crosses the x-axis

Ready to Practice?

Try solving cubic equations with the interactive tool and compare the result with the methods above.

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Find quick answers to common questions about cubic equations and our solving methods.

Still have questions?

What is a cubic equation?

A cubic equation is a third-degree polynomial written in standard cubic form, where the leading coefficient cannot be zero.

Can this solver show complex roots?

Yes. If the equation has one real root and a complex-conjugate pair, the results section shows them clearly and labels them as complex.

Why does coefficient a matter so much?

If a = 0, the equation is no longer cubic. The UI validates this immediately and explains why the solver cannot proceed.

What does the step-by-step section show?

It summarizes the normalized equation, depressed cubic transformation, discriminant, and final interpretation so the solver feels more transparent.