Question

[Trần Minh Ngọc]

Cho \(x=\sqrt[3]{5-\sqrt{17}}+\sqrt[3]{5+\sqrt{17}}\)

CMR : \(x^3-6x-10=0\)

Answer 1

[Thiên thần chính nghĩa]

\(x=\sqrt[3]{5-\sqrt{17}}+\sqrt[3]{5+\sqrt{17}}\)

\(\Rightarrow x^3=5-\sqrt{17}+5+\sqrt{17}+3\sqrt[3]{\left(5-\sqrt{17}\right)\left(5+\sqrt{17}\right)}x\)

\(\Rightarrow x^3=10+3\sqrt[3]{25-17}x\)

\(\Rightarrow x^3=10+3\sqrt[3]{8}x\)

\(\Rightarrow x^3=10+3.2x\)

\(\Leftrightarrow x^3-6x-10=0\)

Học toán vui vẻ!