Question
$$9 = v+\frac{4}{v}+12$$
Answer
$$ \begin{matrix}v_1 = -\dfrac{ 3 }{ 2 }+\dfrac{\sqrt{ 7 }}{ 2 }i & v_2 = -\dfrac{ 3 }{ 2 }-\dfrac{\sqrt{ 7 }}{ 2 }i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} 9 &= v+\frac{4}{v}+12&& \text{multiply ALL terms by } \color{blue}{ v }. \\[1 em]v\cdot9 &= vv+v\cdot\frac{4}{v}+v\cdot12&& \text{cancel out the denominators} \\[1 em]9v &= v^2+4+12v&& \text{simplify right side} \\[1 em]9v &= v^2+12v+4&& \text{move all terms to the left hand side } \\[1 em]9v-v^2-12v-4 &= 0&& \text{simplify left side} \\[1 em]-v^2-3v-4 &= 0&& \\[1 em] \end{aligned} $$
$ -x^{2}-3x-4 = 0 $ is a quadratic equation.
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