Practice Problems

Worked Examples

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

Three Distinct Real Roots

real

x³ - 6x² + 11x - 6 = 0

Solution Steps

Identify: a = 1, b = -6, c = 11, d = -6.

Depress the cubic and obtain p = -1, q = 0.

The discriminant is Delta = -1/27, so there are three real roots.

Apply the trigonometric branch to get x_1 = 1, x_2 = 2, x_3 = 3.

Roots

x₁ = 1 x₂ = 2 x₃ = 3

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

One Real and Two Complex Roots

complex

x³ + x + 2 = 0

Solution Steps

This cubic is already depressed because the x^2 term is missing.

Compute p = 1 and q = 2.

The discriminant is Delta = 28/27, so there is one real root and one complex conjugate pair.

Cardano's formula gives x_1 \approx -1 and the remaining roots are complex.

Roots

x₁ ≈ -1 x₂ ≈ 0.5 + 0.866i x₃ ≈ 0.5 - 0.866i

Verification: f(-1) = -1 + (-1) + 2 = 0

Triple Root

repeated

x³ - 3x² + 3x - 1 = 0

Solution Steps

Recognize the identity (x - 1)^3.

Equivalently, Delta = 0 and the cubic collapses to a triple root.

All three roots are equal to x = 1.

Roots

x = 1 (multiplicity 3)

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

Double Root

repeated

x³ - 5x² + 8x - 4 = 0

Solution Steps

Test rational roots and find x = 1.

Factor the cubic as (x - 1)(x^2 - 4x + 4).

The quadratic factor is (x - 2)^2, so x = 2 is a double root.

Roots

x₁ = 1 x₂ = x₃ = 2

Verification: (x - 1)(x - 2)^2 = x³ - 5x² + 8x - 4

Negative Leading Coefficient

repeated

-2x³ + 6x² - 6x + 2 = 0

Solution Steps

Divide by -2 to normalize the cubic.

The result is x^3 - 3x^2 + 3x - 1 = 0.

That is (x - 1)^3, so the root is x = 1 with multiplicity three.

Roots

x = 1 (multiplicity 3)

Verification: -2(1)^3 + 6(1)^2 - 6(1) + 2 = 0

Pure Cubic

complex

x³ - 8 = 0

Solution Steps

Rearrange to x^3 = 8.

The real cube root gives x = 2.

Factoring gives (x - 2)(x^2 + 2x + 4) = 0, which reveals the complex pair.

Roots

x₁ = 2 x₂ = -1 + i√3 x₃ = -1 - i√3

Verification: 2^3 - 8 = 0

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