Derive the equation of a circle from center/radius and complete the square to identify center/radius.
Complete the square to convert a circle equation to standard form
Problem
Convert \(x^{2}~+~y^{2}~-~6x~+~4y~-~3~=~0\) to center-radius form. Group the \(x\)- and \(y\)-terms, add the \(\text{two}~\text{completing}\text{-}\text{square}~\text{constants}\) to \(\text{both}~\text{sides}\), and factor \(\text{both}~\text{perfect}\text{-}\text{square}~\text{trinomials}\).
Big Picture
What this problem is really about
Completing the square must preserve the equation while turning each variable group into a distance square. We’ll isolate the constant, group x-terms and y-terms, derive each compensation value from half the linear coefficient, add both values to both sides, and factor the two perfect-square trinomials.
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