daca x+[tex] \frac{1}{x} [/tex]=3 |*x
atunci x² +1=3x
x²=3x-1
x²+[tex] \frac{1}{ x^{2} } [/tex] =
= 3x-1+ [tex] \frac{1}{3x-1} [/tex] = [tex] \frac{(3x-1)(3x-1)+1}{(3x-1)} [/tex]=
=[tex] \frac{9 x^{2} -6x+1+1}{(3x-1)} [/tex]=
=[tex] \frac{9*(3x-1)-6x+2}{(3x-1)} [/tex]=
=[tex] \frac{27x-9-6x+2}{(3x-1)} [/tex]=
=[tex] \frac{21x-7}{(3x-1)} [/tex] = [tex] \frac{7*(3x-1)}{(3x-1)} [/tex] = 7
