Question

Calculati expresia (Ex.4 din imagine) si scrieti raspunsul sub forma de fractie ireductibila

Answer 1

[tex]\displaystyle\\ -\frac{x+2}{2-x}+\frac{4x}{x^2-4} -\frac{2-x}{x+2}=~~~~~~~~~~ x \in R-\{-2;~2\} \\\\\ =-\frac{x+2}{-(x-2)}+\frac{4x}{x^2-4}-\frac{-(x-2)}{x+2}=\\\\ =\frac{x+2}{x-2}+\frac{4x}{x^2-4}+\frac{x-2}{x+2}=\\\\ =\frac{(x+2)(x+2)}{(x-2)(x+2)}+\frac{4x}{x^2-4}+\frac{(x-2)(x-2)}{(x+2)(x-2)}=\\ \\ = \frac{x^2+4x+4}{x^2-4}+\frac{4x}{x^2-4}+\frac{x^2-4x+4}{x^2-4}=\\\\ = \frac{x^2+4x+4+4x+x^2-4x+4}{x^2-4}=\\\\ =\frac{2x^2+4x+8}{x^2-4}=\frac{2(x^2+2x+4)}{x^2-4}[/tex]