Roots Of Cubic Equation Calculator

Cubic Equation Formula:

\[ ax^3 + bx^2 + cx + d = 0 \]

Roots:

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1. What is a Cubic Equation?

A cubic equation is a polynomial equation of degree three, with the general form ax³ + bx² + cx + d = 0 where a ≠ 0. Cubic equations always have three roots (solutions), which may be real or complex numbers.

2. How Does the Calculator Work?

The calculator uses the Cardano's method to solve cubic equations:

\[ x = \sqrt[3]{-\frac{g}{2} + \sqrt{h}} + \sqrt[3]{-\frac{g}{2} - \sqrt{h}} - \frac{b}{3a} \] \[ \text{where } f = \frac{\frac{3c}{a} - \frac{b^2}{a^2}}{3}, \quad g = \frac{\frac{2b^3}{a^3} - \frac{9bc}{a^2} + \frac{27d}{a}}{27}, \quad h = \frac{g^2}{4} + \frac{f^3}{27} \]

The discriminant (h) determines the nature of the roots:

h > 0: One real root and two complex conjugate roots
h = 0: All roots real and at least two are equal
h < 0: All roots are real and distinct

3. Types of Roots

Real Roots: When the graph crosses the x-axis. A cubic equation always has at least one real root.

Complex Roots: Occur in conjugate pairs when the discriminant is positive. These don't correspond to x-intercepts on a real graph.

4. Using the Calculator

Tips: Enter coefficients a, b, c, and d. Coefficient 'a' cannot be zero. For best results, use exact coefficients rather than decimal approximations.

5. Frequently Asked Questions (FAQ)

Q1: What if coefficient 'a' is zero?
A: Then it's not a cubic equation. The calculator requires a ≠ 0.

Q2: Why do I get complex roots?
A: Cubic equations with positive discriminant have one real and two complex roots. This is mathematically correct.

Q3: How precise are the results?
A: Results are rounded to 6 decimal places. For exact solutions, symbolic algebra systems may be needed.

Q4: Can I solve higher-order polynomials?
A: This calculator is specifically for cubic equations. Quartics and higher require different methods.

Q5: What about multiple roots?
A: When discriminant is zero, at least two roots are identical. The calculator will show these equal values.