\(\sqrt{4x^2+5x+1}-2\sqrt{x^2-x+1}=9x-3\)
\(\Leftrightarrow\sqrt{4x^2+5x+1}-\dfrac{2\sqrt{7}}{3}-\left(2\sqrt{x^2-x+1}-\dfrac{2\sqrt{7}}{3}\right)=9x-3\)
\(\Leftrightarrow\dfrac{4x^2+5x+1-\dfrac{28}{9}}{\sqrt{4x^2+5x+1}+\dfrac{2\sqrt{7}}{3}}-\dfrac{4\left(x^2-x+1\right)-\dfrac{28}{9}}{2\sqrt{x^2-x+1}+\dfrac{2\sqrt{7}}{3}}=9x-3\)
\(\Leftrightarrow\dfrac{\dfrac{36x^2+45x-19}{9}}{\sqrt{4x^2+5x+1}+\dfrac{2\sqrt{7}}{3}}-\dfrac{\dfrac{36x^2-36x+8}{9}}{2\sqrt{x^2-x+1}+\dfrac{2\sqrt{7}}{3}}=3\left(3x-1\right)\)
\(\Leftrightarrow\dfrac{\dfrac{\left(3x-1\right)\left(12x+19\right)}{9}}{\sqrt{4x^2+5x+1}+\dfrac{2\sqrt{7}}{3}}-\dfrac{\dfrac{4\left(3x-2\right)\left(3x-1\right)}{9}}{2\sqrt{x^2-x+1}+\dfrac{2\sqrt{7}}{3}}-3\left(3x-1\right)=0\)
\(\Leftrightarrow\left(3x-1\right)\left(\dfrac{12x+19}{9\left(\sqrt{4x^2+5x+1}+\dfrac{2\sqrt{7}}{3}\right)}-\dfrac{4\left(3x-2\right)}{9\left(2\sqrt{x^2-x+1}+\dfrac{2\sqrt{7}}{3}\right)}-3\right)=0\)
Dễ thấy: \(\dfrac{12x+19}{9\left(\sqrt{4x^2+5x+1}+\dfrac{2\sqrt{7}}{3}\right)}-\dfrac{4\left(3x-2\right)}{9\left(2\sqrt{x^2-x+1}+\dfrac{2\sqrt{7}}{3}\right)}-3< 0\)
\(\Rightarrow3x-1=0\Rightarrow3x=1\Rightarrow x=\dfrac{1}{3}\)
