Question

Giải phương trình vô tỷ sau:

\(\sqrt[3]{3x^2-x+2001}\) - \(\sqrt[3]{3x^2-7x+2002}\) - \(\sqrt[3]{6x-2003}\) = \(\sqrt[3]{2002}\)

( MN GIÚP MÌNH NHA , MÌNH ĐANG CẦN GẤP )

( CẢM ƠN)

Answer 1

Dùng hđt \(\sqrt[3]{a}-\sqrt[3]{b}=\dfrac{a-b}{\sqrt[3]{a^2}+\sqrt[3]{ab}+\sqrt[3]{b^2}}\) và \(\sqrt[3]{a}+\sqrt[3]{b}=\dfrac{a+b}{\sqrt[3]{a^2}-\sqrt[3]{ab}+\sqrt[3]{b^2}}\)

Ta có:

\(\sqrt[3]{3x^2-x+2001}-\sqrt[3]{3x^2-7x+2002}=\sqrt[3]{6x+2003}+\sqrt[3]{2002}=0\)

\(\Leftrightarrow\dfrac{6x-1}{\sqrt[3]{\left(3x^2-x+2001\right)^2}+\sqrt[3]{\left(3x^2-x+2001\right)\left(3x^2-7x+2002\right)}+\sqrt[3]{\left(3x^2-7x+2002\right)^2}}=\dfrac{6x-1}{\sqrt[3]{\left(6x+2003\right)^2}-\sqrt[3]{2002.\left(6x+2003\right)}+\sqrt[3]{2002^2}}\)

\(\Leftrightarrow x=\dfrac{1}{6}\)