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 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. |
|
