Question

Rút gọn biểu thức: \(\dfrac{x^4y-xy^4}{x^2+xy+y^2}\)

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

b) Tìm x biết: \(\left(x+2\right)^2-\left(x-2\right)\left(x+2\right)=0\)

Answer 1

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

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

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

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

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

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

\(=\dfrac{x^2-y^2}{2\left(x-y\right)\left(x+y\right)}=\dfrac{\left(x-y\right)\left(x+y\right)}{2\left(x-y\right)\left(x+y\right)}=\dfrac{1}{2}\)

b ) \(\left(x+2\right)^2-\left(x-2\right)\left(x+2\right)=0\)

\(\Leftrightarrow\left(x+2\right)\left(x+2-x+2\right)=0\)

\(\Leftrightarrow4\left(x+2\right)=0\)

\(\Leftrightarrow x=2\)