Question

 

Thực hiện phép tính:

\(\frac{2x}{x^2+2xy}+\frac{y}{xy-2y^2}+\frac{4}{x^2-4y^2}\)

 

Answer 1

\(\frac{2x}{x^2+2xy}+\frac{y}{xy-2y^2}+\frac{4}{x^2-4y^2}\)

\(=\frac{2x}{x\left(x+y\right)}+\frac{y}{y\left(x-2y\right)}+\frac{4}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2\left(x-2y\right)+x+2y+4}{\left(x+2y\right)\left(x-2y\right)}\)

\(=\frac{3x-2y+4}{\left(x+2y\right)\left(x-2y\right)}\)

Answer 2

\(ĐKXĐ:\hept{\begin{cases}x\ne0\\y\ne0\\x\ne\pm2y\end{cases}}\)

\(\frac{2x}{x^2+2xy}+\frac{y}{xy-2y^2}+\frac{4}{x^2-4y^2}=\frac{2x}{x\left(x+2y\right)}+\frac{y}{y\left(x-2y\right)}+\frac{4}{\left(x+2y\right)\left(x-2y\right)}\)

\(=\frac{2}{x+2y}+\frac{1}{x-2y}+\frac{4}{\left(x+2y\right)\left(x-2y\right)}\)\(=\frac{2\left(x-2y\right)}{\left(x+2y\right)\left(x-2y\right)}+\frac{x+2y}{\left(x+2y\right)\left(x-2y\right)}+\frac{4}{\left(x+2y\right)\left(x-2y\right)}\)

\(=\frac{2\left(x-2y\right)+x+2y+4}{\left(x+2y\right)\left(x-2y\right)}=\frac{2x-4y+x+2y+4}{\left(x+2y\right)\left(x-2y\right)}\)

\(=\frac{3x-2y+4}{\left(x+2y\right)\left(x-2y\right)}\)

Answer 3

\(\frac{2x}{x^2+2xy}+\frac{y}{xy-2y^2}+\frac{4}{x^2-4y^2}\)

\(=\frac{2x}{x\left(x+2y\right)}+\frac{y}{y\left(x-2y\right)}+\frac{4}{\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2x.y\left(x-2y\right)}{xy\left(x-2y\right)\left(x+2y\right)}+\frac{y.x\left(x+2y\right)}{xy\left(x-2y\right)\left(x+2y\right)}+\frac{4xy}{xy\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2xy\left(x-2y\right)+xy\left(x+2y\right)+4xy}{xy\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{2x^2-4xy^2+x^2y+2xy^2+4xy}{xy\left(x-2y\right)\left(x+2y\right)}\)

\(=\frac{3x^2y-2xy^2+4xy}{xy\left(x-2y\right)\left(x+2y\right)}\)

P/s:#Học Tốt#