Cho x=[tex]\sqrt[3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}[/tex]
CMR với mọi a>[tex]\frac{1}{8}[/tex] thì x là số nguyên dương
Bằng tất cả sự chân thành mong các bạn giúp đỡ!!!
[tex]x=\sqrt[3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}[/tex]
[tex]\Rightarrow x^3=2a+3.\sqrt[3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}.\sqrt[3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}(\sqrt[3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}})[/tex]
[tex]\Leftrightarrow x^3=2a.\sqrt[3]{(a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}})(a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}})}x[/tex]
[tex]\Leftrightarrow x^3=2a+3\sqrt[3]{a^2-(\frac{a+1}{3})^2.(\sqrt{\frac{8a-1}{3}})^2}.x[/tex]
[tex]\Leftrightarrow x^3=2a+3\sqrt[3]{\frac{(1-2a)^3}{3^3}}x[/tex]
[tex]\Leftrightarrow x^3=2a+(1-2a)x[/tex]
[tex]\Leftrightarrow x^3=2a-2ax+x[/tex]
[tex]\Leftrightarrow x^3-x-2a+2ax=0\Leftrightarrow x(x-1)(x+1)+2a(x-1)=0\Leftrightarrow (x-1)(x^2+x+2a)=0\Rightarrow \begin{bmatrix} x-1=0 & & \\ x^2+x+2a=0 & & \end{bmatrix}\Leftrightarrow \begin{bmatrix} x=1 & & \\ VN_0 & & \end{bmatrix}[/tex]
Vậy có $dpcm$
[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
