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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
The formed product, a carbon radical, can react with non-radical molecule to continue propagation or react with another radical to form a new stable molecule such as a longer carbon chain or an alkyl halide. [4] The example below of methane chlorination shows a multi-step reaction involving radicals.
The final step is called termination (6,7), in which the radical recombines with another radical species. If the reaction is not terminated, but instead the radical group(s) go on to react further, the steps where new radicals are formed and then react are collectively known as propagation (4,5). This is because a new radical is created, able ...
Radical extensions occur naturally when solving polynomial equations in radicals. In fact a solution in radicals is the expression of the solution as an element of a radical series: a polynomial f over a field K is said to be solvable by radicals if there is a splitting field of f over K contained in a radical extension of K.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
When the task is to find the solution that is the best under some criterion, this is an optimization problem. Solving an optimization problem is generally not referred to as "equation solving", as, generally, solving methods start from a particular solution for finding a better solution, and repeating the process until finding eventually the ...
Explicit and implicit methods — implicit methods need to solve an equation at every step; Backward Euler method — implicit variant of the Euler method; Trapezoidal rule — second-order implicit method; Runge–Kutta methods — one of the two main classes of methods for initial-value problems Midpoint method — a second-order method with ...
is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals. The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the quadratic formula, the cubic formula, and the quartic formula for degrees two, three, and ...