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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 ...
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.
Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation. The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.
(square roots are nonnegative by definition of the notation). As the inequality may always be satisfied by possibly exchanging x and y, solving the first equation in x and y is equivalent with solving + = +.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
To complete the square, form a squared binomial on the left-hand side of a quadratic equation, from which the solution can be found by taking the square root of both sides. The standard way to derive the quadratic formula is to apply the method of completing the square to the generic quadratic equation a x 2 + b x + c = 0 {\displaystyle ...
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 by the equation, consisting generally of two expressions related by an equals sign.
Finding the root of a linear polynomial (degree one) is easy and needs only one division: the general equation + = has solution = /. For quadratic polynomials (degree two), the quadratic formula produces a solution, but its numerical evaluation may require some care for ensuring numerical stability.
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