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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.
Because of this, often, the only simple effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation. After discarding solutions that yield an invalid equation, we will have the correct set of solutions.
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 ...
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.
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 ...
Thus solving a polynomial system over a number field is reduced to solving another system over the rational numbers. For example, if a system contains 2 {\displaystyle {\sqrt {2}}} , a system over the rational numbers is obtained by adding the equation r 2 2 – 2 = 0 and replacing 2 {\displaystyle {\sqrt {2}}} by r 2 in the other equations.
A reaction step involving two molecular entities is called bimolecular. A reaction step involving three molecular entities is called trimolecular or termolecular. In general, reaction steps involving more than three molecular entities do not occur, because is statistically improbable in terms of Maxwell distribution to find such a transition state.