$$x+\frac{2}{x}+\frac{4}{3}x = 11$$
Answer
$$ \begin{matrix}x_1 = \dfrac{ 33 }{ 14 }-\dfrac{\sqrt{ 921 }}{ 14 } & x_2 = \dfrac{ 33 }{ 14 }+\dfrac{\sqrt{ 921 }}{ 14 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x+\frac{2}{x}+\frac{4}{3}x &= 11&& \text{multiply ALL terms by } \color{blue}{ x\cdot3 }. \\[1 em]x\cdot3x+x\cdot3\cdot\frac{2}{x}+x\cdot3\frac{4}{3}x &= x\cdot3\cdot11&& \text{cancel out the denominators} \\[1 em]3x^2+6+4x^2 &= 33x&& \text{simplify left side} \\[1 em]7x^2+6 &= 33x&& \text{move all terms to the left hand side } \\[1 em]7x^2+6-33x &= 0&& \text{simplify left side} \\[1 em]7x^2-33x+6 &= 0&& \\[1 em] \end{aligned} $$
$ 7x^{2}-33x+6 = 0 $ is a quadratic equation.
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