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A quadratic integer is an algebraic integer of degree two. More explicitly, it is a complex number = /, which solves an equation of the form x 2 + bx + c = 0, with b and c integers.
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 ...
For quadratic equations with rational coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrationals. If the discriminant is zero, then there is exactly one real root − b 2 a , {\displaystyle -{\frac {b}{2a}},} sometimes called a repeated or double root or two equal roots.
Otherwise, the quadratic polynomial cannot be factorized into non-constant real factors. The quadratic formula is valid when the coefficients belong to any field of characteristic different from two, and, in particular, for coefficients in a finite field with an odd number of elements. [9]
The study of quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares , which determines when an integer may be expressed in the form x 2 + y 2 , where x , y are integers.
The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number =. There are analogous polynomials for p = 2 , 3 , 5 , 11 and 17 {\displaystyle p=2,3,5,11{\text{ and }}17} (the lucky numbers of Euler ), corresponding to other Heegner numbers.
A quadratic form with integer coefficients is called an integral binary quadratic form, often abbreviated to binary quadratic form. This article is entirely devoted to integral binary quadratic forms. This choice is motivated by their status as the driving force behind the development of algebraic number theory.
which is a simple quadratic equation, whose solutions are easily found using the quadratic formula: ... are three different real numbers or is a real number and and ...