Question

Решить уравнение во вложении

Answer 1

$x^6=\frac{257x^2-68}{68x^2-257}; x^2=t\ge 0; t^3=\frac{257t-68}{68t-257}; t\not= \frac{257}{68}; t^3(68t-257)=257t-68;$

$68t^4-257t^3-257t+68=0;\ t\not= 0;\ 68(t^2+\frac{1}{t^2})-257(t+\frac{1}{t})=0;\ t+\frac{1}{t}=p>0;$

$68(p^2-2)-257p=0;\ 68p^2-257p-136=0; D=257^2-4\cdot 68\cdot (-136)=321^2;$

$p=\frac{257\pm 321}{2\cdot 68}=\left [ {{17/4} \atop {-8/17<0}} \right. ; t+\frac{1}{t}=\frac{17}{4}; \left [ {{t=4} \atop {t=1/4}} \right. ; \left [ {{x=\pm 2} \atop {x=\pm 1/2}} \right.$

Ответ: $\pm 2; \pm\frac{1}{2}$