Question

Solve the following quadratic equations by factorization:(x+3/x-2)-(1-x/x)=17/4

Answer 1

$\text{Consider,}$

$\displaystyle\frac{x+3}{x-2}-\frac{1-x}{x}=\frac{17}{4}$

$\displaystyle\frac{x(x+3)-(1-x)(x-2)}{(x-2)x}=\frac{17}{4}$

$\displaystyle\frac{x^2+3x-(x-2-x^2+2x)}{(x-2)x}=\frac{17}{4}$

$\displaystyle\frac{x^2+3x-(-x^2+3x-2)}{x^2-2x}=\frac{17}{4}$

$\displaystyle\frac{x^2+3x+x^2-3x+2}{x^2-2x}=\frac{17}{4}$

$\displaystyle\frac{2x^2+2}{x^2-2x}=\frac{17}{4}$

$\displaystyle\;8x^2+8=17x^2-34x$

$\displaystyle\;9x^2-34x-8=0$

$\displaystyle\;9x^2-36x+2x-8=0$

$\displaystyle\;9x(x-4)+2(x-4)=0$

$\displaystyle\;(x-4)(9x+2)=0$

$\implies\bf\displaystyle\;x=4,\frac{-2}{9}$

$\therefore\textbf{The solution set is }\{\bf\;4,\frac{-2}{9}\}$