Question
$$x+\frac{7}{x}+17 = \frac{3}{2}$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 31 }{ 4 }-\dfrac{\sqrt{ 849 }}{ 4 } & x_2 = -\dfrac{ 31 }{ 4 }+\dfrac{\sqrt{ 849 }}{ 4 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} x+\frac{7}{x}+17 &= \frac{3}{2}&& \text{multiply ALL terms by } \color{blue}{ x\cdot2 }. \\[1 em]x\cdot2x+x\cdot2\cdot\frac{7}{x}+x\cdot2\cdot17 &= x\cdot2\cdot\frac{3}{2}&& \text{cancel out the denominators} \\[1 em]2x^2+14+34x &= 3x&& \text{simplify left side} \\[1 em]2x^2+34x+14 &= 3x&& \text{move all terms to the left hand side } \\[1 em]2x^2+34x+14-3x &= 0&& \text{simplify left side} \\[1 em]2x^2+31x+14 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}+31x+14 = 0 $ is a quadratic equation.
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