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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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