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A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots. When there is only one distinct root, it can be interpreted as two roots with the same value, called a double root. When there are no real roots, the coefficients can be considered as complex numbers with ...
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.
Real-root isolation is useful because usual root-finding algorithms for computing the real roots of a polynomial may produce some real roots, but, cannot generally certify having found all real roots. In particular, if such an algorithm does not find any root, one does not know whether it is because there is no real root.
If >, the cubic has three distinct real roots; If <, the cubic has one real root and two non-real complex conjugate roots. This can be proved as follows. First, if r is a root of a polynomial with real coefficients, then its complex conjugate is also a root. So the non-real roots, if any, occur as pairs of complex conjugate roots.
A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, + and , where i is the imaginary unit. In general, any non-zero complex number has n distinct complex-valued n th roots, equally distributed around a complex circle of constant absolute value .
In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots. More generally, the discriminant of a univariate polynomial of positive degree is zero if and only if the polynomial has a multiple root.
The roots may be real or complex, as well as distinct or repeated. If a characteristic equation has parts with distinct real roots, h repeated roots, or k complex roots corresponding to general solutions of y D (x), y R 1 (x), ..., y R h (x), and y C 1 (x), ..., y C k (x), respectively, then the general solution to the differential equation is
Casus irreducibilis occurs when none of the roots are rational and when all three roots are distinct and real; the case of three distinct real roots occurs if and only if q 2 / 4 + p 3 / 27 < 0, in which case Cardano's formula involves first taking the square root of a negative number, which is imaginary, and then taking the ...