Question

\(x=\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\)

Answer 1

\(x=\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\)

<=>\(x^3=5+2\sqrt{13}+3.\sqrt[3]{5+2\sqrt{13}}.\sqrt[3]{5-2\sqrt{13}}\left(\sqrt[3]{5-2\sqrt{13}}+\sqrt[3]{5+2\sqrt{13}}\right)+5-2\sqrt{13}\)

<=> \(x^3=10+3\sqrt[3]{5^2-\left(2\sqrt{13}\right)^2}.x\)

<=> \(x^3=10+3\sqrt[3]{-27}.x=10-9x\)

<=> x³+9x-10=0

<=> x³-x²+x²-x+10x-10=0

<=>\(x^2\left(x-1\right)+x\left(x-1\right)+10\left(x-1\right)=0\)

<=> \(\left(x^2+x+10\right)\left(x-1\right)=0\)

<=> \(\left(x^2+2.\frac{1}{2}x+\frac{1}{4}+\frac{39}{4}\right)\left(x-1\right)=0\)

<=> \(\left[\left(x+\frac{1}{2}\right)^2+\frac{39}{4}\right]\left(x-1\right)=0\)

=> x-1=0 (vì \(\left(x+\frac{1}{2}\right)^2+\frac{39}{4}>0\))

<=> x=1