Cubic Equation Calculator

A standard cubic equation is expressed in the polynomial form: ax³ + bx² + cx + d = 0, where a cannot equal zero. In our calculation engine, these coefficients correspond to A3, A2, A1, and A0 respectively. To determine the nature of the roots, the calculator first evaluates the cubic discriminant using the coefficients. If the discriminant is positive, the equation yields one distinct real root and two complex conjugate roots. If it equals zero, all roots are real and at least two are equal. If it is negative, there are three distinct real roots. The tool then applies advanced numerical methods and analytical formulas to extract Root 1, Root 2, and Root 3 with your desired numerical precision.

Let us solve the cubic equation: x³ - 6x² + 11x - 6 = 0. Here, our coefficients are A3 = 1, A2 = -6, A1 = 11, and A0 = -6. First, we look for an initial rational root using the Rational Root Theorem, testing factors of the constant term (-6). Trying x = 1: (1)³ - 6(1)² + 11(1) - 6 = 1 - 6 + 11 - 6 = 0. Since the result is zero, x = 1 is our first root. Next, we use polynomial long division or synthetic division to divide our cubic polynomial by (x - 1), yielding the quadratic equation: x² - 5x + 6 = 0. Factoring this quadratic gives (x - 2)(x - 3) = 0. Therefore, our three exact roots are x = 1, x = 2, and x = 3.

When working with cubic equations, always ensure your equation is written in standard descending order of exponents before identifying your coefficients. If your leading coefficient (A3) is negative, you can multiply the entire equation by -1 to simplify manual factoring attempts. Additionally, pay close attention to complex roots; if your discriminant indicates complex numbers, they will always appear in conjugate pairs.