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Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
The quadratic formula is exactly correct when performed using the idealized arithmetic of real numbers, but when approximate arithmetic is used instead, for example pen-and-paper arithmetic carried out to a fixed number of decimal places or the floating-point binary arithmetic available on computers, the limitations of the number representation ...
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula. A quadratic polynomial or quadratic function can involve ...
All quadratic equations have exactly two solutions in complex numbers (but they may be equal to each other), a category that includes real numbers, imaginary numbers, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation
Equations can be classified according to the types of operations and quantities involved. Important types include: An algebraic equation or polynomial equation is an equation in which both sides are polynomials (see also system of polynomial equations). These are further classified by degree: linear equation for degree one; quadratic equation ...
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Equally important as the use or lack of symbolism in algebra was the degree of the equations that were addressed. Quadratic equations played an important role in early algebra; and throughout most of history, until the early modern period, all quadratic equations were classified as belonging to one of three categories.
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