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\begin{align}  &\iff (f^{2}+g^{2})+(fd-eg)+b=0\\  &\iff f^{2}+fd+g^{2}-eg+b=0\\  &\iff (f+\frac{1}{2}d)^{2}-\frac{1}{4}d^{2}+(g-\frac{1}{2})^{2}-\frac{1}{4}e^{2}+b=0 \text{ (By completing the square)} square)}\\  &\iff (f+\frac{1}{2}d)^{2}+(g-\frac{1}{2}e)^2=\frac{1}{4}d^{2}+\frac{1}{4}e^{2}-b\\  &\iff(f+\frac{1}{2}d)^{2}+(g-\frac{1}{2}e)^2=\frac{1}{4}(d^{2}+e^{2})-b\\  \end{align}