Question

$8 {x}^{3} - 8 {x}^{2} + 1 = 0$

Answer 1

$8x^3-8x^2+1=0;\ (2x)^3-2(2x)^2+1=0;\ 2x=t;\ t^3-2t^2+1=0;\ (t^3-t^2)-(t^2-1)=0;\ t^2(t-1)-(t-1)(t+1)=0;\ (t-1)(t^2-t-1)=0;$

1) t-1=0; t=1; x=1/2

2) $t^2-t-1=0;\ t=\frac{1\pm\sqrt{5}}{2};\ x=\frac{1\pm \sqrt{5}}{4}$

Ответ: $\{\frac{1}{2};\ \frac{1\pm\sqrt{5}}{4}\}$