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A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
This happens to coincide with the "Babylonian" method of finding square roots, which consists of replacing an approximate root x n by the arithmetic mean of x n and a ⁄ x n. By performing this iteration, it is possible to evaluate a square root to any desired accuracy by only using the basic arithmetic operations.
Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian clay tablets dating from 1800–1600 BCE, and is still taught in elementary algebra courses today.
Graeffe's method – Algorithm for finding polynomial roots; Lill's method – Graphical method for the real roots of a polynomial; MPSolve – Software for approximating the roots of a polynomial with arbitrarily high precision; Multiplicity (mathematics) – Number of times an object must be counted for making true a general formula
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
The Galois theory approach to analyzing and solving polynomials is to ask whether, given coefficients of a polynomial each of which is a symmetric function in the roots, one can "break" the symmetry and thereby recover the roots. Using this approach, solving a polynomial of degree is related to the ways of rearranging ("permuting ...
The modular square root of can be taken this way. Having solved the associated quadratic equation we now have the variables w and set v = r (if C in the quadratic is a natural square). Solve for variables α {\displaystyle \alpha } and β {\displaystyle \beta } the following equation:
Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. [16] [17] Linearization is another technique for solving nonlinear equations.