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Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC. [4] [5] Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation.
The problem of constructing a regular pentagon is equivalent to the problem of constructing the roots of the equation z 5 − 1 = 0. One root of this equation is z 0 = 1 which corresponds to the point P 0 (1, 0). Removing the factor corresponding to this root, the other roots turn out to be roots of the equation z 4 + z 3 + z 2 + z + 1 = 0.
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
When the monic quadratic equation with real coefficients is of the form x 2 = c, the general solution described above is useless because division by zero is not well defined. As long as c is positive, though, it is always possible to transform the equation by subtracting a perfect square from both sides and proceeding along the lines ...
If each unknown appears at most twice, then a word equation is called quadratic; in a quadratic word equation the graph obtained by repeatedly applying Levi's lemma is finite, so it is decidable if a quadratic word equation has a solution. [2] A more general method for solving word equations is Makanin's algorithm. [3] [4]
The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]