Question

Thực hiện phép tính: \(\dfrac{x}{{x + y}} + \dfrac{{2xy}}{{{x^2} - {y^2}}} - \dfrac{y}{{x + y}}\) 

Answer 1

`x/(x+y) + (2xy)/(x^2-y^2) - y(x+y)`

`= (x(x-y))/(x^2-y^2) + (2xy)/(x^2-y^2) - (y(x-y))/(x^2-y^2)`

`= (x^2 - xy + 2xy - xy + y^2)/(x^2-y^2)`

`= (x^2+y^2)/(x^2-y^2)`

Answer 2

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

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

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

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

\(=\dfrac{x^2+y^2}{x^2-y^2}\)

Answer 3

\(MTC:x^2-y^2=\left(x+y\right)\left(x-y\right)\\ =\dfrac{x\left(x-y\right)}{\left(x-y\right)\left(x+y\right)}+\dfrac{2xy}{x^2-y^2}-\dfrac{y\left(x-y\right)}{\left(x+y\right)\left(x-y\right)}\\ =\dfrac{x\left(x-y\right)+2xy-y\left(x-y\right)}{x^2-y^2}\\ =\dfrac{x^2-xy+2xy-xy+y^2}{x^2-y^2}=\dfrac{x^2+y^2}{x^2-y^2}\)