# 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 isnt 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 .
| * 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 7*x *= 35* *has the single root *x *= 5* . *Raising this equation to the second power, we receive the equation:
49*x*^{2} = 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* *49*x*^{2} = 1225 results in 7*x *= 35 , and we lose the root: *x *=* *5. A *right* extraction of this root leads to the equation: | 7*x *| = 35,* *hence the two cases imply:
1) * *7*x *= 35,* *then* x *=* *5 ;* *2) * *7*x *= 35,* * then* x *=* *5 .
Hence, at a *right *extraction of square root we dont 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. | Back |