The equation is called irrational if some algebraic rational expression from the unknown is under the sign of the radical. At the solution of the irrational equations the task of finding only of the valid roots is set.
Instruction
1. Any irrational equation, it is possible to present in the form of the algebraic equation which will be a consequence of initial. Transformations, such as multiplication of both parts by the same expression containing unknown, transferring composed from one part in another, reduction of similar and removal of a multiplier for brackets and also construction of both members of equation in the whole positive degree are for this purpose used.
2. At the same time it must be kept in mind that the rational equation received thus can be nonequivalent to the initial irrational equation and contain excess roots which will not be roots of this irrational equation. In this regard, all received roots of the rational algebraic equation, it is necessary to check by substitution in the initial equation whether for the purpose of examination they are roots of the irrational equation.
3. A main goal when transforming the irrational equations, is receiving not just any algebraic rational equation, and receiving the equation formed of polynomials as is possible for smaller degree, having solved which, you will find also roots of the initial equation.
4. The easiest way of the solution of the irrational equation is application of a method of release from radicals. It consists in consecutive construction of the left and right member of equation in the corresponding natural degree. Using this method it is necessary to remember that at construction in even degree, the received equation will be nonequivalent initial, and if in odd, then the equivalent equation will turn out. Despite such lack of this method, it is the most widespread.
5. The second method of the solution of the irrational equations consists in introduction of new unknown that the initial equation or to simpler irrational, or leads to the rational equation.