Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Complete the square and report center and radius separately
Problem
For \(x^{2}~+~y^{2}~-~6x~+~4y~-~3~=~0\), group terms and complete \(\text{both}~\text{squares}\) while keeping the equation balanced. Report the resulting center-radius form, then read the center and positive radius separately.
Big Picture
What this problem is really about
This conversion has two stages: first create center-radius form without changing the equation, then decode its geometric parameters. We’ll complete and balance both variable groups, factor the squares, reverse the signs inside the binomials for the center, and take the nonnegative square root of the right side for the radius.
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