Quadratic equation/Advanced

In mathematics, or more specifically algebra, a quadratic equation is one involving only polynomials of the second degree. Quadratic equations are a common part of mathematical solutions to real-world problems in a huge variety of situations. Fortunately, there exists a simple closed formula for finding the roots of such an equation, the quadratic formula.

Quadratic equations occurring in applications typically involve real number coefficients. However, one can manipulate polynomials in the usual way as long as the coefficients can be added and multiplied together. Please see the main page for a discussion of polynomials with real coefficients.

The most general mathematical context that deals with systems of objects that can be added and multiplied together is ring (mathematics) theory. One can work with polynomials, and in particular quadratic polynomial equations, as long as the coefficients are in a ring. The real numbers is an example of a ring. Another example, important in coding theory, is polynomials with coefficients in the ring $$\mathbb{Z}_2 = \{ \, \overline{0}, \overline{1} \, \}$$. You add and multiply in this ring in the same way you add or multiply the integers $$\{ \, 0, 1, \, \}$$ with one exception: since $$\mathbb{Z}_2$$ does not have a "two" in it, we set $$\overline{1} + \overline{1} = \overline{0}$$.

Solutions of quadratic equations
When working with polynomials over a specific ring $$R$$, one usually looks for solutions in the same ring $$R$$. The main exception to this is the most common case, where a polynomial has integer coefficients but one desires real number solutions. If, instead, one demands solutions of the same type as the polynomial coefficients, namely integers, the equation becomes a Diophantine equation. In this article, we assume that the desired solutions are in the same ring that the coefficients are drawn from.

Every polynomial equation with coefficients in a ring $$R$$ can be put into the form:


 * $$ax^2+bx+c=0\,$$

with a, b and c in $$R$$ and $$a\not=0$$. When the coefficients are real numbers, the quadratic formula specifies the roots of this equation as


 * $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\ .$$

If $$R$$ is an arbitrary ring, however, there are several problems with this formula. The derivation of the quadratic formula typically involves completing the square.