We know that the fraction is equal to zero in that case when its numerator is equal to zero, i.e. x ² + x-2=0, and the denominator is not equal Solving the equation x ² + x-2=0, we find roots h1=1, h2 = – But number 1 does not enter ODZ of this equation that is, the initial equation has one root x =-
Let a h=a – an equation root. Means numerical equality of f (a) of =g (a) +q (a) takes place. But then on property of real numbers also numerical equality of f (a) - q (a) of =g (a) showing that and – an equation root will be carried out (. It is similarly proved that each root of the equation (is also an equation root (.
So, the equations which contain a variable under the sign of a root, are called irrational. The irrational equations are solved generally construction of both members of equation in a square (or n-uyu degree) or introduction of a new variable. Besides, also solutions of the irrational equations use artificial methods.
So, if at the solution of the equation there is a transition to the equation – to a consequence, there could be foreign roots. In this case all roots of the equation consequence need to be checked, substituting them in the initial equation. In certain cases identification of foreign roots is facilitated by knowledge of ODZ of the initial equation – the roots which are not belonging to ODZ can be rejected at once. So, in the given example the foreign root h=1 does not enter equation ODZ
At the solution of the equations various identical transformations over the expressions entering the equation are carried out. Thus the initial equation changes others, having the same roots. Such equations are called equivalent.
Definition: The equation of f (x) =g(x) is equivalent to the equation of f1 (x) =g1 (x) if each root of the first equation is a root of the second and back, each root of the second equation is a root of the first, i.e. their decision coincide.