Worked Examples
Master cubic equations through practice. Each example is solved step by step and presented in clean textbook notation.
1
examples_ex1_title
Real x³ - 6x² + 11x - 6 = 0
Solution Steps
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examples_ex1_step1
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examples_ex1_step2
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examples_ex1_step3
4
examples_ex1_step4
examples_final_roots
x₁ = 1 x₂ = 2 x₃ = 3
Verification: (x - 1)(x - 2)(x - 3) = x³ - 6x² + 11x - 6
2
examples_ex2_title
Repeated x³ + 3x² - 4 = 0
Solution Steps
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examples_ex2_step1
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examples_ex2_step2
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examples_ex2_step3
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examples_ex2_step4
examples_final_roots
x₁ = 1 x₂ = -2 (repeated)
Verification: (x - 1)(x + 2)² = x³ + 3x² - 4
3
examples_ex3_title
Real x³ - x = 0
Solution Steps
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examples_ex3_step1
2
examples_ex3_step2
3
examples_ex3_step3
examples_final_roots
x₁ = 0 x₂ = 1 x₃ = -1
Verification: x(x - 1)(x + 1) = x³ - x
examples_faq_title
Find quick answers to common questions about cubic equations and our solving methods.
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.