Question

Solve x minus 1 divided by x minus 2 minus x minus 3 / x - 4 is equal to 10 by 3

Answer 1

$\textbf{Given:}$

$\displaystyle\frac{x-1}{x-2}-\frac{x-3}{x-4}=\frac{10}{3}$

$\displaystyle\frac{(x-1)(x-4)-(x-2)(x-3)}{(x-2)(x-4)}=\frac{10}{3}$

$\displaystyle\frac{(x^2-5x+4)-(x^2-5x+6)}{x^2-6x+8}=\frac{10}{3}$

$\displaystyle\frac{-2}{x^2-6x+8}=\frac{10}{3}$

$\displaystyle\frac{-1}{x^2-6x+8}=\frac{5}{3}$

$\displaystyle\,-3=5x^2-30x+40$

$\implies\displaystyle\bf\,5x^2-30x+43=0$

$\text{We cannot solve this quadratic equation by factorization method }$

$\text{So, we apply formula method}$

$\displaystyle\bf\,x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

$\displaystyle\,x=\frac{30\pm\sqrt{30^2-4(5)(43)}}{2(5)}$

$\displaystyle\,x=\frac{30\pm\sqrt{900-860}}{10}$

$\displaystyle\,x=\frac{30\pm\sqrt{40}}{10}$

$\displaystyle\,x=\frac{30\pm2\sqrt{10}}{10}$

$\displaystyle\bf\,x=\frac{15\pm\sqrt{10}}{5}$

$\therefore\textsf{Solution set is}\{\frac{15+\sqrt{10}}{5},\;{\frac{15-\sqrt{10}}{5}\}}$