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Finding roots in a specific region of the complex plane, typically the real roots or the real roots in a given interval (for example, when roots represents a physical quantity, only the real positive ones are interesting). For finding one root, Newton's method and other general iterative methods work generally well.
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
A quadratic function without real root: y = (x − 5) 2 + 9. The "3" is the imaginary part of the x-intercept. The real part is the x-coordinate of the vertex. Thus the roots are 5 ± 3i. The solutions of the quadratic equation + + = may be deduced from the graph of the quadratic function = + +, which is a parabola.
where C is a constant approximately equal to 0.625 735 8072. [23] In other words, the expected number of real roots of a random polynomial of high degree is lower than the natural logarithm of the degree.
Denoting the two roots by r 1 and r 2 we distinguish three cases. If the discriminant is zero the fraction converges to the single root of multiplicity two. If the discriminant is not zero, and |r 1 | ≠ |r 2 |, the continued fraction converges to the root of maximum modulus (i.e., to the root with the greater absolute value).
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction. For example, 3 is a square root of 9, since 3 2 = 9, and −3 is also a square root of 9, since (−3) 2 = 9.
The square root of a univariate quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola. If a > 0 , {\displaystyle a>0,} then the equation y = ± a x 2 + b x + c {\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}} describes a hyperbola, as can be seen by squaring both sides.
For three values of n, the roots of unity are quadratic integers: For n = 3, 6 they are Eisenstein integers (D = −3). For n = 4 they are Gaussian integers (D = −1): see Imaginary unit. For four other values of n, the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its complex conjugate (also an n ...