Geometry

Start by moving all the terms to one side and combining like terms:

\(4x^2-8x+4y^2+14y+12=0\)

Then, divide both sides by \(4 \):

\(x^2-2x+y^2+\frac{7}{2}y+3=0\)

Next, complete the square for \(x \) and \(y \):

\((x-1)^2+(y+\frac{7}{4})^2-1-\frac{49}{16}+3=0\)

Simplify:

\((x-1)^2+(y+\frac{7}{4})^2=\frac{17}{16}\)

This is the equation (in standard form) for the circle. Now take the square root of \(\frac{17}{16} \):

\(\sqrt{\frac{17}{16}}=\frac{\sqrt{17}}{\sqrt{16}}=\boxed{\frac{\sqrt{17}}{4}}\)  Which is our answer.