trial-and-error
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analytical method.
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They both give you exactly the same value. However, in some cases solving a problem using both methods is very difficult.
Methods vary considerably depending upon the number of powers in the equation. For example, the method for solving cubics is quite different to solving quadratics etc... It's not really possible to generalise to one technique.
The graphical method is a method used to solve algebraical problems by using graphs.