Question

Решите уравнение (x-4)/(x+2)+(x+2)/(x-4)=17/4

Answer 1

$\frac{x-4}{x+2}+ \frac{x+2}{x-4}= \frac{17}{4} \\ \frac{(x-4)^{2}+(x+2)^{2}}{(x+2)(x-4)}- \frac{17}{4} =0\\ \frac{4(x-4)^{2}+4(x+2)^{2}-17(x+2)(x-4)}{4(x+2)(x-4)} =0\\ \frac{(2*(x-4))^{2}-2*2*2*(x+2)(x-4)+(2*(x+2))^{2}-9(x+2)(x-4)}{4(x+2)(x-4)} =0\\ \frac{(2(x-4)-2(x+2))^{2}-9(x+2)(x-4)}{4(x+2)(x-4)} =0\\ \frac{(2x-8-2x-4))^{2}-9(x^{2}-2x-8)}{4(x+2)(x-4)} =0\\ \frac{144-9x^{2}+18x+72)}{4(x+2)(x-4)} =0\\ x^{2}-2x-24=0 \\ x \neq -2 \\ x \neq 4 \\ D=4+96=100 \\ x_{1}=\frac{2+10}{2}=6 \\$
$x_{2}=\frac{2-10}{2}=-4 \\$
ответ: 6; -4