Question

CMR \(x_0=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\) la nghiem cua phuong trinh 

\(\left(x^3-3x-17\right)^{2017}-1=0\)

Answer 1

\(A=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\)

\(\Leftrightarrow A^3=9+4\sqrt{5}+9-4\sqrt{5}+3\cdot A\cdot\sqrt[3]{\left(9+4\sqrt{5}\right)\left(9-4\sqrt{5}\right)}\)

\(\Leftrightarrow A^3=18+3\cdot A\cdot1=18+3A\)

\(\Leftrightarrow A^3-3A-18=0\)

=>A=3

\(\left(x^3-3x-17\right)^{2017}-1=0\)

\(\Leftrightarrow x^3-3x-17=1\)

\(\Leftrightarrow x^3-3x-18=0\)

\(\Leftrightarrow x^3-3x^2+3x^2-9x+6x-18=0\)

=>(x-3)(x²+3x+6)=0

=>x=3