\(\Leftrightarrow8x^3-36x^2+51x-22+2x-3-\sqrt[3]{3x-5}=0\)
\(\Leftrightarrow8x^3-36x^2+51x-22+\dfrac{8x^3-36x^2+51x-22}{\left(2x-3\right)^2+\left(2x-3\right)\sqrt[3]{3x-5}+\sqrt[3]{\left(3x-5\right)^2}}=0\)
\(\Leftrightarrow\left(8x^3-36x^2+51x-22\right)\left(1+\dfrac{1}{\left(2x-3\right)^2+\left(2x-3\right)\sqrt[3]{3x-5}+\sqrt[3]{\left(3x-5\right)^2}}\right)=0\)
\(\Leftrightarrow8x^3-36x^2+51x-22=0\)
\(\Leftrightarrow\left(x-2\right)\left(8x^2-20x+11\right)=0\)
\(\Leftrightarrow...\)
Cách khác: (Đưa về hàm đặc trưng)
\(PT\Leftrightarrow8x^3-36x^2+53x-25=\sqrt[3]{3x-5}\)
\(\Leftrightarrow\left(2x-3\right)^3+2x-3=3x-5+\sqrt[3]{3x-5}\). (*)
Xét hàm \(f\left(t\right)=t^3+t\). Ta thấy f(t) đồng biến trên \(\mathbb{R}\).
Do đó \(\left(\cdot\right)\Leftrightarrow2x-3=\sqrt[3]{3x-5}\)
\(\Leftrightarrow8x^3-36x^2+54x-27=3x-5\)
\(\Leftrightarrow8x^3-36x^2+51x-22=0\)
\(\Leftrightarrow\left(x-2\right)\left(8x^2-20x+11\right)=0\Leftrightarrow...\)
