[tex] \sqrt[3]{x^{3}-2 x^{2} +2x+7} = x+1[/tex] ⇔
⇔ [tex]x^{3}-2 x^{2} +2x+7 = (x+1)^{3} [/tex] ⇔
[tex] (x+y)^{3}= x^{3} +3 x^{2} y+3x y^{2} + y^{3} [/tex] (formula)
⇔ [tex]x^{3}-2 x^{2} +2x+7=x^{3}+3 x^{2} +3x+1 [/tex] ⇔
⇔ [tex]x^{3}-2 x^{2} +2x+7-x^{3}-3 x^{2} -3x-1=0[/tex] ⇔
⇔ [tex]-5 x^{2} -x+6=0[/tex]
Δ=[tex] b^{2} -4ac= (-1)^{2} -4*(-5)*6=1+120=121[/tex]
[tex] x_{1}= \frac{1+ 11 }{2*(-5)}= \frac{12}{-10} =- \frac{6}{5}
[/tex]
[tex] x_{2} = \frac{1-11}{2*(-5)}= \frac{-10}{-10}=1 [/tex] ⇒
⇒ [tex] x_{1}= -\frac{6}{5} [/tex] si [tex] x_{2} = 1[/tex].