Calculate a Cubic Equation

Calculator for cubic equations. The three results of the equation ax³+bx²+cx+d=0 will be calculated. Such a third-degree equation always has at least one real solution, but the other two solutions can also be complex numbers. Please enter a value for each coefficient a, b c and d. If this is omitted, then a is set to 1, the others to 0. The formula is converted to normal form so that the parameter a of x³ equals 1. This normal form is a different function than the one given, but has the same zeros.
Not for every equation a solution is found!

The solution formulas for cubic equations of the form x³+a₂x²+a₁x+a₀=0 are:
a=12a₁³-3a₁²a₂²-54a₁a₂a₀+81a₀²+12a₀a₂³
b=³√36a₁a₂-108a₀-8a₂³+12√a
x₁=(b²-12a₁+4a₂²-2a₂b)/(6b)
p=(-b²+12a₁-4a₂²-4a₂b)/(-6b)
q=(a₁a₂b-9ba₀+b√a+b²a₁-2a₁a₂²-18a₂a₀+2a₂√a+12a₁²)/(3b²)
x₂=-p/2+i√-(p/2)²+q
x₃=-p/2-i√-(p/2)²+q

The graph of a cube function is continuous, always comes from the direction of minus or plus infinity and goes in the opposite direction. If a is positive, then it goes from minus to plus infinity, otherwise the other way. Therefore, the graph must intersect the x-axis and therefore has at least one real zero. This applies to all odd-numbered polynomials. Even-numbered polynomials, on the other hand, can have real zeros, but they don't have to, since they go either from infinity to infinity or from negative infinity to negative infinity and therefore don't have to intersect the x-axis.

The ancient Babylonians discovered how to solve quadratic equations over 4,000 years ago. However, this was not known for cubic equations for a long time. It wasn't until 1545 that the Italian mathematician Gerolamo Cardano published his solution formula for reduced cubic equations of the form x³+px+q=0, i.e., without x². The above solution formulas can be derived from Cardano's formulas.