Question

Consideram expresia E(x) = ( 2 / x-2 + x / x+2 ) : x^2 + 4 / x^2 - x - 2 unde x este un numarul real, x nu = -2 , x nu = -1 si x nu = 2
Aratati ca E(x) = x+1 / x+2 pentru orice x numar real , x nu = -2, x nu = -1 si x nu = 2

/ - linie de fractie
x^2 - x la a doua

Answer 1

Răspuns:

$E(x)=(\frac{2}{x-2}+\frac{x}{x+2}):\frac{x^{2}+4}{x^{2}-x-2}\\ \\=\frac{2(x+2)+x(x-2)}{(x+2)(x-2)}*\frac{x^{2}-x-2}{x^{2}+4}\\ \\=\frac{x^{2}+4}{(x+2)(x-2)}*\frac{x^{2}+x-2x-2}{x^{2}+4}\\ \\=\frac{x(x+1)-2(x+1)}{(x+2)(x-2)}\\\\=\frac{(x-2)(x+1)}{(x+2)(x-2)}\\ \\=\frac{x+1}{x+2}$