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Pell's equation. Pell's equation for n = 2 and six of its integer solutions. Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form where n is a given positive nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions ...
Continued fraction. A finite regular continued fraction, where is a non-negative integer, is an integer, and is a positive integer, for . In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this ...
Partial fraction decomposition. In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions ...
With this framework we apply the cover-up rule to solve for A, B, and C. D 1 is x + 1; set it equal to zero. This gives the residue for A when x = −1. Next, substitute this value of x into the fractional expression, but without D 1. Put this value down as the value of A. Proceed similarly for B and C. D 2 is x + 2; For the residue B use x = −2.
Equation solving. The quadratic formula, the symbolic solution of the quadratic equation ax2 + bx + c = 0. An example of using Newton–Raphson method to solve numerically the equation f(x) = 0. In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated ...
1 + x 1 − 1 = 0. Since P 1 (x) < 0 for x = 1 / 2 , and P 1 (x) > 0 for all x > 1, the root of P 1 lies between 1 / 2 and 1, and the first term in its greedy expansion (the second term in the greedy expansion for the golden ratio) is 1 / 2 . If x 2 is the remaining fraction after this step of the greedy expansion, it ...
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [1][2] For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously ...
Quadratic formula. The roots of the quadratic function y = 1 2 x2 − 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.