Worked Examples

Master cubic equations through practice. Each example is solved step by step and presented in clean textbook notation.

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Real

x³ - 6x² + 11x - 6 = 0

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x₁ = 1 x₂ = 2 x₃ = 3

Verification: (x - 1)(x - 2)(x - 3) = x³ - 6x² + 11x - 6

Repeated

x³ + 3x² - 4 = 0

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x₁ = 1 x₂ = -2 (repeated)

Verification: (x - 1)(x + 2)² = x³ + 3x² - 4

Real

x³ - x = 0

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x₁ = 0 x₂ = 1 x₃ = -1

Verification: x(x - 1)(x + 1) = x³ - x

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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.