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As with all kinds of problem solving, trial and error may sometimes yield a solution, in particular where the form of the equation, or its similarity to another equation with a known solution, may lead to an "inspired guess" at the solution.
However note that in attempting to find solutions for this equation, if we modify the function's definition – more specifically, the function's domain, we can find solutions to this equation.
So, if we were instead to define that the domain of ƒ consists of the real numbers, the equation above has two solutions, and its solution set is (with a, b, c, d, and k real-valued constants) is a hyperplane.
The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all equations or inequalities.
If the solution set is empty, then there are no values x such that the equations or inequalities becomes true simultaneously.
A solution is an assignment of expressions to the unknown variables that makes the equality in the equation true.
In other words, a solution is an expression or a collection of expressions (one for each unknown) such that, when substituted for the unknowns, the equation becomes an identity.
Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example , which simplifies this to a quadratic equation in z).
In Diophantine equations the solutions are required to be integers.
Depending on the problem, the task may be to find any solution (finding a single solution is enough) or all solutions.
The set of all solutions is called the solution set. It is also possible that the task is to find a solution, among possibly many, that is best in some respect; problems of that nature are called optimization problems; solving an optimization problem is generally not referred to as "equation solving".