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.