Expresie evaluarea nationala 2019

Inainte de a aduce la acelasi numitor comun, nu e rau sa vedem daca nu putem simplifica fractiile. Daca am studia doar primul factor al expresiei am avea surprize placute. In afara valorilor critice de la numitorii fractiilor, usor de identificat, avem:

$$\frac{x^2-x}{(x-1)(x-3)}-\frac{3}{x-3}-\frac{x}{x+1}=\frac{x(x-1)}{(x-1)(x-3)}-\frac{3}{x-3}-\frac{x}{x+1}=\frac{x}{x-3}-\frac{3}{x-3}-\frac{x}{x+1}=$$
$$=\frac{x-3}{x-3}-\frac{x}{x+1}=1-\frac{x}{x+1}=\frac{1}{x+1}$$
$$E(x)=\frac{1}{x+1}:\frac{x-1}{x^2-1}=\frac{1}{x+1}.\frac{(x-1)(x+1)}{x-1}=1$$

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