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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 ( 3x+ 2 )2 = 15x + 10  may be replaced by the next equivalent equation:  9x2 + 12x + 4 = 15x + 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":  9x2+ 12x + 4 – 15x – 10 = 0, after this we receive: 9x2 – 3x – 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:

3x2 –  x – 2 = 0 .

This equation is equivalent to an original one:

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

   4. It is possible to raise both sides of an equation to an odd power and to extract the odd degree root from both sides of an equation.
It is necessary to remember that:
         a)  raising to an even power can result in acquisition of extraneous roots;
         b)  a wrong extraction of even degree root can result in loss of roots.

E x a m p l e s :  The equation  7x = 35  has the single root  x = 5 . Raising this equation to
                          the second power, we receive the equation:

49x2 = 1225 ,

                          having the two roots:  x = 5  and  x = – 5 . The last value is an extraneous root. A wrong
                          extraction of square root from both sides of the equation  49x2 = 1225 results in 7x = 35 ,
                          and we lose the root:  x = – 5. A right extraction of this root leads to the equation:
                           | 7x | = 35,  hence the two cases imply:

                           1)   7x = 35,  then  x = 5 ;        2)   – 7x = 35,  then  x = – 5 .

                          Hence, at a right extraction of square root we don’t  lose roots of an equation. What means
                          a right extraction of a root ? Here we meet the notion of an arithmetical root, which is
                          considered further in the section of the same name.

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