Cubic Equation Solver
Polynomial equations in the form
Ax3 + Bx2 + Cx + D = 0
are called cubic equations. If the coefficient A is not equal to zero, then you can divide both sides of the equation by A and simplify the equation to
x3 + bx2 + cx + d = 0.
If a cubic equation has coefficients that are real numbers, then it has three roots, at least one of which is a real number. The other two roots may be a repeated real root, or a pair of complex conjugates. In every case, the values of the roots can be determined by the coefficients of the polynomial. The formula that gives the roots in terms of the coefficients is called the Cubic Formula. It is similar the Quadratic Formula for quadratic equations, except that it is much more complicated. You can find the explicit formulas here, or use the cubic equation calculator on the left.
Here are examples of cubic equations with real, repeated, and complex roots:
x3 - 6x2 + 11x - 6 = 0; solutions: x = 1, 2, 3
x3 + x2 - 16x + 20 = 0; solutions: x = -5, 2, 2
x3 + 6x2 + 12x + 8 = 0; solutions: x = -2, -2, -2
x3 - 6x2 + 10x - 8 = 0; solutions: x = 4, 1 + i, 1 - i
x3 + 9x2 + 9x + 81 = 0; solutions: x = -9, 3i, -3i
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