Main ways used at solving of equations
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:
|
This equation is equivalent to an original one: