Main ways used at solving of equations
Transferring terms of equation from one side to another.
Multiplication and division by nonzero 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 righthand 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:

This equation is equivalent to an original one: