# Main ways used at solving of equations

Identical transformations. Replacement of expression.
Transferring terms of equation from one side to another.
Multiplication and division by non-zero expression (number).
Raising to a power. Extraneous roots of equation.
Extracting of a root. Loss of roots of equation.

Solving of equation is a process, consisting mainly in a replacement of the given equation by another, equivalent equation. This replacement is called an identical transformation . Main identical transformations are the following.

 1 Replacement of one expression by another, identically equal to it. For example, the equation ( 3 x+ 2 ) 2 = 15 x + 10  may be replaced by the next equivalent equation:  9 x 2 + 12 x + 4 = 15 x + 10 . 2 Transferring terms of equation from one side to another with back signs. So, in the previous equation we can transfer all terms from the right-hand side to the left with the sign "minus": 9 x 2 + 12 x + 4 – 15 x – 10 = 0, after this we receive: 9 x 2 – 3 x – 6 = 0 . 3 Multiplication or division of both sides of equation by the same expression ( number ), not equal to zero. This is very important, because a new equation can be not equivalent to previous, if the expression, by which we multiply or divide, can be equal to zero. E x a m p l e :  The equation x – 1 = 0 has the single root x = 1 . Multiplying it by x – 3 , we receive the equation ( x – 1 )( x – 3 ) = 0, which has two roots: x = 1 and x = 3 . The last value isn’t a root for the given equation x – 1 = 0 . This value is so called an extraneous root. And vice versa, division can result to a loss of roots . In our case, if  ( x – 1 )( x – 3 ) = 0 is the origin equation, then the root x = 3 will be lost at division of this equation by x – 3 . In the last equation (p.2) we can divide all terms by 3 (not zero!) and finally receive: 3 x 2 –  x – 2 = 0 .

This equation is equivalent to an original one:

( 3 x+ 2 ) 2 = 15 x + 10 .