Bài 1: Thực hiện phép tính (quy đồng)
1) \(\frac{1}{y(x-y)} - \frac{1}{x(x-y)}\)
2) \(\frac{x+6}{2x+6} + \frac{2x+3}{x(x+3)}\)
3) \(\frac{x}{5x+5} - \frac{x}{10x-10}\)
4) \(\frac{1}{xy-x^2} - \frac{1}{y^2-xy}\)
5) \(\frac{1}{y^2-xy} + \frac{1}{x^2-xy}\)
6) \(\frac{x}{y^2-xy} - \frac{y}{xy-x^2}\)
7) \(\frac{x+3}{x^2-1} - \frac{x+1}{x^2-x}\)
8) \(\frac{x+9}{x^2-9} - \frac{3}{x^2+3x}\)
9) \(\frac{x-12}{6x-36} + \frac{6}{x^2-6x}\)
10) \(\frac{3x+5}{x^2-5x} + \frac{25-x}{25-5x}\)
11) \(\frac{3}{2x+6} - \frac{x-6}{2x^2+6x}\)
12) \(\frac{y}{2x^2-xy} + \frac{4x}{y^2-2xy}\)
13) \(\frac{1}{x-5x^2} - \frac{25x-15}{25x^2-1}\)
14) \(\frac{x+1}{2x-2} + \frac{x^2+3}{2-2x^2}\)
15) \(\frac{7}{x} - \frac{x}{x+6} + \frac{36}{x^2+6x}\)
16) \(\frac{3}{2x} + \frac{3x-3}{2x-1} + \frac{2x^2+1}{4x^2-2x}\)
17) \(\frac{1}{x} + \frac{
1: \(\dfrac{1}{y\left(x-y\right)}-\dfrac{1}{x\left(x-y\right)}=\dfrac{x-y}{xy\left(x-y\right)}=\dfrac{1}{xy}\)
2: \(\dfrac{x+6}{2x+6}+\dfrac{2x+3}{x\left(x+3\right)}\)
\(=\dfrac{x\left(x+6\right)+2\left(2x+3\right)}{2x\left(x+3\right)}\)
\(=\dfrac{x^2+6x+4x+6}{2x\left(x+3\right)}=\dfrac{x^2+10x+6}{2x\left(x+3\right)}\)
3: \(\dfrac{x}{5x+5}-\dfrac{x}{10x-10}\)
\(=\dfrac{x}{5\left(x+1\right)}-\dfrac{x}{10\left(x-1\right)}\)
\(=\dfrac{2x\left(x-1\right)-x\left(x+1\right)}{10\left(x+1\right)\left(x-1\right)}\)
\(=\dfrac{2x^2-2x-x^2-x}{10\left(x+1\right)\left(x-1\right)}=\dfrac{x^2-3x}{10\left(x+1\right)\left(x-1\right)}\)
4: \(\dfrac{1}{xy-x^2}-\dfrac{1}{y^2-xy}\)
\(=\dfrac{1}{x\left(y-x\right)}-\dfrac{1}{y\left(y-x\right)}=\dfrac{y-x}{xy\left(y-x\right)}=\dfrac{1}{xy}\)
5: \(\dfrac{1}{y^2-xy}+\dfrac{1}{x^2-xy}\)
\(=\dfrac{1}{y\left(y-x\right)}+\dfrac{1}{x\left(x-y\right)}\)
\(=\dfrac{x-y}{xy\left(y-x\right)}=\dfrac{-1}{xy}\)
6: \(\dfrac{x}{y^2-xy}-\dfrac{y}{xy-x^2}\)
\(=\dfrac{x}{y\left(y-x\right)}-\dfrac{y}{x\left(y-x\right)}\)
\(=\dfrac{x^2-y^2}{xy\left(y-x\right)}=\dfrac{-\left(y-x\right)\left(y+x\right)}{xy\left(y-x\right)}=\dfrac{-\left(y+x\right)}{xy}\)
7: \(\dfrac{x+3}{x^2-1}-\dfrac{x+1}{x^2-x}\)
\(=\dfrac{x+3}{\left(x-1\right)\left(x+1\right)}-\dfrac{x+1}{x\left(x-1\right)}\)
\(=\dfrac{x\left(x+3\right)-\left(x+1\right)^2}{x\left(x-1\right)\left(x+1\right)}=\dfrac{x^2+3x-x^2-2x-1}{\left(x-1\right)\cdot\left(x+1\right)\cdot x}\)
\(=\dfrac{x-1}{x\left(x-1\right)\left(x+1\right)}=\dfrac{1}{x\left(x+1\right)}\)
8: \(\dfrac{x+9}{x^2-9}-\dfrac{3}{x^2+3x}\)
\(=\dfrac{x+9}{\left(x-3\right)\left(x+3\right)}-\dfrac{3}{x\left(x+3\right)}\)
\(=\dfrac{x\left(x+9\right)-3\left(x-3\right)}{x\left(x+3\right)\left(x-3\right)}=\dfrac{x^2+9x-3x+9}{x\left(x+3\right)\left(x-3\right)}\)
\(=\dfrac{\left(x+3\right)^2}{x\left(x+3\right)\left(x-3\right)}=\dfrac{x+3}{x\left(x-3\right)}\)
9: \(\dfrac{x-12}{6x-36}+\dfrac{6}{x^2-6x}\)
\(=\dfrac{x-12}{6\left(x-6\right)}+\dfrac{6}{x\left(x-6\right)}\)
\(=\dfrac{x\left(x-12\right)+36}{6x\left(x-6\right)}=\dfrac{x^2-12x+36}{6x\left(x-6\right)}\)
\(=\dfrac{\left(x-6\right)^2}{6x\left(x-6\right)}=\dfrac{x-6}{6x}\)
10: \(\dfrac{3x+5}{x^2-5x}+\dfrac{25-x}{25-5x}\)
\(=\dfrac{3x+5}{x\left(x-5\right)}+\dfrac{x-25}{5\left(x-5\right)}\)
\(=\dfrac{5\left(3x+5\right)+x\left(x-25\right)}{5x\left(x-5\right)}\)
\(=\dfrac{15x+25+x^2-25x}{5x\left(x-5\right)}=\dfrac{x^2-10x+25}{5x\left(x-5\right)}\)
\(=\dfrac{\left(x-5\right)^2}{5x\left(x-5\right)}=\dfrac{x-5}{5x}\)
11: \(\dfrac{3}{2x+6}-\dfrac{x-6}{2x^2+6x}\)
\(=\dfrac{3}{2\left(x+3\right)}-\dfrac{x-6}{2x\left(x+3\right)}\)
\(=\dfrac{3x-x+6}{2x\left(x+3\right)}=\dfrac{2x+6}{2x\left(x+3\right)}=\dfrac{1}{x}\)
12: \(\dfrac{y}{2x^2-xy}+\dfrac{4x}{y^2-2xy}\)
\(=\dfrac{y}{x\left(2x-y\right)}+\dfrac{4x}{y\left(y-2x\right)}\)
\(=\dfrac{y}{x\left(2x-y\right)}-\dfrac{4x}{y\left(2x-y\right)}\)
\(=\dfrac{y^2-4x^2}{xy\left(2x-y\right)}=\dfrac{\left(y-2x\right)\left(y+2x\right)}{-xy\left(y-2x\right)}=\dfrac{y+2x}{-xy}\)
13: \(\dfrac{1}{x-5x^2}-\dfrac{25x-15}{25x^2-1}\)
\(=\dfrac{-1}{x\left(5x-1\right)}-\dfrac{25x-15}{\left(5x-1\right)\left(5x+1\right)}\)
\(=\dfrac{-5x-1-x\left(25x-15\right)}{x\left(5x-1\right)\left(5x+1\right)}\)
\(=\dfrac{-5x-1-25x^2+15x}{x\left(5x-1\right)\left(5x+1\right)}=\dfrac{-25x^2+10x-1}{x\left(5x-1\right)\left(5x+1\right)}\)
\(=\dfrac{-\left(5x-1\right)^2}{x\left(5x-1\right)\left(5x+1\right)}=\dfrac{-5x+1}{x\left(5x+1\right)}\)
14: \(\dfrac{x+1}{2x-2}+\dfrac{x^2+3}{2-2x^2}\)
\(=\dfrac{x+1}{2\left(x-1\right)}-\dfrac{x^2+3}{2\left(x-1\right)\left(x+1\right)}\)
\(=\dfrac{\left(x+1\right)^2-x^2-3}{2\left(x-1\right)\left(x+1\right)}=\dfrac{2x-2}{2\left(x-1\right)\left(x+1\right)}=\dfrac{1}{x+1}\)
15: \(\dfrac{7}{x}-\dfrac{x}{x+6}+\dfrac{36}{x^2+6x}\)
\(=\dfrac{7}{x}-\dfrac{x}{x+6}+\dfrac{36}{x\left(x+6\right)}\)
\(=\dfrac{7\left(x+6\right)-x^2+36}{x\left(x+6\right)}=\dfrac{7\left(x+6\right)-\left(x-6\right)\left(x+6\right)}{x\left(x+6\right)}\)
\(=\dfrac{7-x+6}{x}=\dfrac{13-x}{x}\)
16: \(\dfrac{3}{2x}+\dfrac{3x-3}{2x-1}+\dfrac{2x^2+1}{4x^2-2x}\)
\(=\dfrac{3\left(2x-1\right)+2x\left(3x-3\right)+2x^2+1}{2x\left(2x-1\right)}\)
\(=\dfrac{6x-3+6x^2-6x+2x^2+1}{2x\left(2x-1\right)}=\dfrac{8x^2-2}{2x\left(2x-1\right)}\)
\(=\dfrac{2\left(4x^2-1\right)}{2x\left(2x-1\right)}=\dfrac{\left(2x-1\right)\left(2x+1\right)}{x\left(2x-1\right)}=\dfrac{2x+1}{x}\)
17: \(\dfrac{1}{x}+\dfrac{1}{x+5}+\dfrac{x-5}{x^2+5x}\)
\(=\dfrac{1}{x}+\dfrac{1}{x+5}+\dfrac{x-5}{x\left(x+5\right)}\)
\(=\dfrac{x+5+x+x-5}{x\left(x+5\right)}=\dfrac{3x}{x\left(x+5\right)}=\dfrac{3}{x+5}\)
8: \(\dfrac{x}{x^2-5x+6}-\dfrac{2}{2-x}+\dfrac{x}{x-3}\)
\(=\dfrac{x}{\left(x-2\right)\left(x-3\right)}+\dfrac{2}{x-2}+\dfrac{x}{x-3}\)
\(=\dfrac{x+2\left(x-3\right)+x\left(x-2\right)}{\left(x-3\right)\left(x-2\right)}\)
\(=\dfrac{x+2x-6+x^2-2x}{\left(x-3\right)\left(x-2\right)}=\dfrac{x^2+x-6}{\left(x-3\right)\left(x-2\right)}\)
\(=\dfrac{\left(x+3\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)}=\dfrac{x+3}{x-3}\)
19: \(\dfrac{x-4}{x^2-2x}+\dfrac{1}{x-2}-\dfrac{1}{x}\)
\(=\dfrac{x-4}{x\left(x-2\right)}+\dfrac{1}{x-2}-\dfrac{1}{x}\)
\(=\dfrac{x-4+x-\left(x-2\right)}{x\left(x-2\right)}=\dfrac{2x-4-x+2}{x\left(x-2\right)}=\dfrac{x-2}{x\left(x-2\right)}=\dfrac{1}{x}\)
20: \(\dfrac{1-3x}{2x}+\dfrac{3x-2}{2x-1}+\dfrac{3x-2}{2x-4x^2}\)
\(=\dfrac{-3x+1}{2x}+\dfrac{3x-2}{2x-1}-\dfrac{3x-2}{2x\left(2x-1\right)}\)
\(=\dfrac{\left(-3x+1\right)\left(2x-1\right)+2x\left(3x-2\right)-3x+2}{2x\left(2x-1\right)}\)
\(=\dfrac{-6x^2+3x+2x-1+6x^2-4x-3x+2}{2x\left(2x-1\right)}\)
\(=\dfrac{-2x+1}{2x\left(2x-1\right)}=-\dfrac{1}{2x}\)
