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.