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If the coefficients , , and are real numbers then when > , the equation has two distinct real roots; when = , the equation has one repeated real root; and when < , the equation has no real roots but has two distinct complex roots, which are complex conjugates of each other.
A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation [ 3 ] a x 2 + b x + c = a ( x − r ) ( x − s ) = 0 {\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0} where r and s are the solutions for x .
In a field of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any. Given an odd prime number p, let q = p e for some positive integer e. A non-zero element of the field F q with q elements is a quadratic residue if it has a square root in F q. Otherwise, it is a quadratic non-residue.
The discriminant is zero if and only if at least two roots are equal. If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers, and negative if there is one real root and two complex conjugate roots. [9]
In mathematics, a quadratic function of a single variable is a function of the form [1] = + +,,where is its variable, and , , and are coefficients.The expression + + , especially when treated as an object in itself rather than as a function, is a quadratic polynomial, a polynomial of degree two.
There are two complex square roots of −1: i and −i, just as there are two complex square roots of every real number other than zero (which has one double square root). In contexts in which use of the letter i is ambiguous or problematic, the letter j is sometimes used instead.
Any nth degree polynomial has exactly n roots in the complex plane, if counted according to multiplicity. So if f(x) is a polynomial with real coefficients which does not have a root at 0 (that is a polynomial with a nonzero constant term) then the minimum number of nonreal roots is equal to (+),
The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. [3]