How to find the solution set?

To find the solution set of an equation, or to solve the equation, one can apply different exact or approximate methods. There is no single method for all types of equations.

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A solution set is the set of solutions of some equation. For an equation with one variable, `f ( x ) = 0 , ` this set is usually called the set of roots. An equation may also involve two or more variables—for example, `x^2+y^2+z^2 = 1 .`

Some types of equations admit exact methods of solving. For example, a linear equation `a x + b = 0 ` has the only solution `x = - b / a ` (for `a != 0 `), (i.e., its solution set consists of one point). Also, one can exactly solve a quadratic equation `a x^2 + bx + c = 0 ` using the quadratic formula `x_(1,2) = ( -b +- sqrt( b^2 - 4ac )) / ( 2a ) .`

Some functions were "invented" as solutions of equations, for example roots of n-th degree and inverse trigonometric functions.

There are also some methods to solve equations approximately. If a real-valued function `f(x) ` is continuous on `[ a, b ] ` and has different signs on its endpoints, i.e. `f(a) lt 0, f(b)gt0 ` or vice versa, then there is at least one root on `(a,b). ` One can test the midpoint of this interval and either spot a root or narrow the interval where it must be. Enough such steps give a root with the desired accuracy.

Some equations have no roots, so their solution sets are empty.

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