جواب معادله در خود معادله صدق میکند با جایگذاری $x = - 3$ در معادله داریم:
$\frac{{x - 3}}{{x - 2}} - \frac{x}{k} = \frac{k}{{x(x + 2)}} \to x = - 3 \to \frac{{ - 3 - 3}}{{ - 3 - 2}} - \frac{{( - 3)}}{k} = \frac{k}{{ - 3( - 3 + 2)}}$
$\frac{6}{5} + \frac{3}{k} = \frac{k}{3} \Rightarrow \frac{k}{3} - \frac{3}{k} - \frac{6}{5} = 0 \to $
$\frac{{6{k^2}}}{{15k}} - \frac{{45}}{{15k}} - \frac{{18k}}{{15k}} = 0 \Rightarrow \frac{{5{k^2} - 18k - 45}}{{15k}} = 0$
حال معادلهٔ صورت عبارت گویا را حل میکنیم:
$5{k^2} - 18k - 45 = 0 \to a{x^2} + bx + c = 0$
$\eqalign{
& \to a = 5 \cr
& b = - 18 \cr
& c = - 45 \cr} $
$\Delta = {b^2} - 4ac \Rightarrow \Delta = {( - 18)^2} - 4 \times (5) \times ( - 45) = 1224$
${x_1} = \frac{{ - b + \sqrt \Delta }}{{2a}} \Rightarrow {x_1} = \frac{{ - ( - 18) + \sqrt {1224} }}{{2 \times 5}} = \frac{{18 + 6\sqrt {34} }}{{10}} = 0/6(3 + \sqrt {34} )$
${x_2} = \frac{{ - b - \sqrt \Delta }}{{2a}} \Rightarrow {x_2} = \frac{{ - ( - 18) - \sqrt {1224} }}{{2 \times 5}} = \frac{{18 - 6\sqrt {34} }}{{10}} = 0/6(3 - \sqrt {34} )$