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Complete Guide

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

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FAQ: Solving Cubic Equations

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

Still have questions?

What is the fastest way to solve a cubic equation?

First try factoring and the rational root theorem. If those fail, reduce the equation to depressed form and use Cardano's method.

Can all cubic equations be solved algebraically?

Yes. Every cubic equation has a closed-form solution using radicals, although some cases are easier to solve with trigonometric or numerical methods.

What if there is no rational root?

If the rational root theorem does not find a root, the cubic may be irreducible over the rationals. In that case, Cardano's method or numerical approximation is the next step.

What is casus irreducibilis?

It is the case where all three roots are real but Cardano's radical form passes through complex numbers. The trigonometric method resolves that case cleanly.