$$\frac{200}{x}-\frac{200}{1.6}x+\frac{1}{2} = 1$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 800 }-\dfrac{\sqrt{ 1024001 }}{ 800 } & x_2 = -\dfrac{ 1 }{ 800 }+\dfrac{\sqrt{ 1024001 }}{ 800 } \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{200}{x}-\frac{200}{1.6}x+\frac{1}{2} &= 1&& \text{multiply ALL terms by } \color{blue}{ x\cdot1.6\cdot2 }. \\[1 em]x\cdot1.6\cdot2\cdot\frac{200}{x}-x\cdot1.6\cdot2\frac{200}{1.6}x+x\cdot1.6\cdot2\cdot\frac{1}{2} &= x\cdot1.6\cdot2\cdot1&& \text{cancel out the denominators} \\[1 em]640-400x^2+x &= 2x&& \text{simplify left side} \\[1 em]-400x^2+x+640 &= 2x&& \text{move all terms to the left hand side } \\[1 em]-400x^2+x+640-2x &= 0&& \text{simplify left side} \\[1 em]-400x^2-x+640 &= 0&& \\[1 em] \end{aligned} $$
$ -400x^{2}-x+640 = 0 $ is a quadratic equation.
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