Question

[Mai Trần]

2. Thực hiện phép tính:

a) \(\frac{1}{2x-3}-\frac{1}{2x+3}\)

b) \(\frac{1}{xy-x^2}-\frac{1}{y^2-xy}\)

c) \(\frac{x+1}{x+4}-\frac{x^2-4}{x^2-16}\)

d)\(\frac{x+1}{2x+6}+\frac{2x+3}{x\left(x+3\right)}\)

e) \(\frac{1-3x}{2x}+\frac{3x-2}{2x-1}+\frac{3x-2}{2x-4x^2}\)

Answer 1

[Nguyễn Lê Phước Thịnh]

a: \(\dfrac{1}{2x-3}-\dfrac{1}{2x+3}=\dfrac{2x+3-2x+3}{\left(2x-3\right)\left(2x+3\right)}=\dfrac{6}{4x^2-9}\)

b: \(\dfrac{1}{xy-x^2}-\dfrac{1}{y^2-xy}\)

\(=\dfrac{1}{x\left(y-x\right)}-\dfrac{1}{y\left(y-x\right)}=\dfrac{y-x}{xy\left(y-x\right)}=\dfrac{1}{xy}\)

c: \(\dfrac{x+1}{x+4}-\dfrac{x^2-4}{x^2-16}\)

\(=\dfrac{x+1}{x+4}-\dfrac{x^2-4}{\left(x-4\right)\left(x+4\right)}\)

\(=\dfrac{\left(x+1\right)\left(x-4\right)-x^2+4}{\left(x+4\right)\left(x-4\right)}=\dfrac{x^2-3x-4-x^2+4}{\left(x+4\right)\left(x-4\right)}=\dfrac{-3x}{x^2-16}\)

d: \(\dfrac{x+1}{2x+6}+\dfrac{2x+3}{x\left(x+3\right)}\)

\(=\dfrac{x+1}{2\left(x+3\right)}+\dfrac{2x+3}{x\left(x+3\right)}\)

\(=\dfrac{x\left(x+1\right)+2\left(2x+3\right)}{2x\left(x+3\right)}=\dfrac{x^2+x+4x+6}{2x\left(x+3\right)}\)

\(=\dfrac{x^2+5x+6}{2x\left(x+3\right)}=\dfrac{\left(x+3\right)\left(x+2\right)}{2x\left(x+3\right)}=\dfrac{x+2}{2x}\)

e: \(\dfrac{1-3x}{2x}+\dfrac{3x-2}{2x-1}+\dfrac{3x-2}{2x-4x^2}\)

\(=\dfrac{-3x+1}{2x}+\dfrac{3x-2}{2x-1}-\dfrac{3x-2}{2x\left(2x-1\right)}\)

\(=\dfrac{\left(-3x+1\right)\left(2x-1\right)+2x\left(3x-2\right)-3x+2}{2x\left(2x-1\right)}\)

\(=\dfrac{-6x^2+3x+2x-1+6x^2-4x-3x+2}{2x\left(2x-1\right)}\)

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