Equations where the zeros of a polynomial are searched for are called algebraic equations. The best-known equations of this type are:
The degree of the equation, i.e. the highest occurring exponent of the unknown x, is essential for the difficulty of an algebraic equation. Linear and quadratic equations were solved quite early, for example by Babylonian, Greek, Indian and Arabic mathematicians. With 3rd and 4th degree equations, success was not achieved until the 16th century. At that time, the Italian mathematicians Scipione del Ferro, Nicolo Tartaglia, Gerolamo Cardano and Lodovico Ferrari developed methods for solving such equations.
To solve the equations mentioned so far, only the four basic arithmetic operations as well as square roots and cube roots are required. The search for similar formulas to solve equations of 5th and higher degree remained unsuccessful despite intensive efforts. In 1824, the Norwegian Niels Henrik Abel was finally able to prove that no general solution formulas can be formulated for such equations using basic arithmetic operations and roots.
The handling of algebraic equations becomes relatively transparent when complex numbers are used. The number of complex solutions is equal to the degree of the equation. However, solutions can coincide. You may enter complex numbers (for example 1.5, −2i or 3+4i) as coefficients in the input fields of this calculator. After clicking on the OK button or pressing the Enter key, the equation (with the unknown z) and the approximated solutions will be displayed on the left-hand side.