Main ways used at solving of equationsIdentical 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 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: 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 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.