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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.
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 zero imaginary part, and the quadratic equation still has two complex-valued roots, complex conjugates of each-other with a non-zero ...
Repeated division of an interval in two results in a set of disjoint intervals, each containing one root, and together listing all the roots. This approach is used in the fastest algorithms today for computer computation of real roots of polynomials (see real-root isolation).
A quadratic polynomial with two real roots (crossings of the x axis). The graph of a real single-variable quadratic function is a parabola. If a quadratic function is equated with zero, then the result is a quadratic equation.
It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. [2] This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them; But a polynomial of odd degree has an odd number of roots;
A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions. For complex roots, one also needs to find a series of similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case the root path will ...
Each root can be approximated by (for example) ... two or more real roots inside the interval (0, 1). In this case cut it in half and consider separately the roots ...
In the case in which the cubic has only one real root, the real root is given by this expression with the radicands of the cube roots being real and with the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to ...