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Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
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 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 ...
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 rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ratio is from unity, the more quickly the continued fraction converges. When the monic quadratic equation with real coefficients is of the form x 2 = c, the general solution described above is useless because division by zero is not well ...
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. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic ...
Muller's method is a recursive method that generates a new approximation of a root ξ of f at each iteration using the three prior iterations. Starting with three initial values x 0, x −1 and x −2, the first iteration calculates an approximation x 1 using those three, the second iteration calculates an approximation x 2 using x 1, x 0 and x −1, the third iteration calculates an ...
It is also used for graphing quadratic functions, deriving the quadratic formula, and more generally in computations involving quadratic polynomials, for example in calculus evaluating Gaussian integrals with a linear term in the exponent, [2] and finding Laplace transforms. [3] [4]
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