Notion of the Consequence of Equation. Extraneous Roots.

Suppose we are given two equations `f_1(x)=g_1(x)` and `f_2(x)=g_2(x)`.

If every root of first equation is also root of the second equation, then second equation is called consequence of the first equation.

Note, that equivalence of equations means that every equation is consequence of another equation.

In process of solution we often need to apply such transformations, that lead to the equation that is consequence of the initial one. All roots of the initial equation satisfy equation-consequence, but the equation-consequence can have some roots that don't satisfy initial equation. Such roots are called extraneous roots. To find extraneous roots and eliminate them we ususally do the following: all roots of the equation-consequence are checked by the substituting into initial equation.

In case when initial equation is substituted by equation-consequence, checking should be performed always.

Therefore, it is important to know what manipulations transform initial equation into equation-consequence:

  1. If both sides of equation are multiplied by expression, that makes sense for all x, then we will obtain equation that is consequence of the initial one. For example `x(x+2)=0` is consequence of `x+2=0` (after multiplying by x). Initial equation has root -2, while equation consequence has two roots: 0 and -2.
  2. If we square both sides of the equation (or raise to some even power), then we will obtain equation that is consequence of the initial one. For example, `x^2=9` is consequence of `x=3` (after squaring both sides). Equation consequence has two roots: 3 and -3, while initial equation has only one root: 3.
    However, if we raise both sides of equation to some odd power, then we will obtain equivalent equation.