Explanation

Expanded form of a circle comes from expanding the standard equation:

(x – h)^2 + (y – k)^2 = r^2 —-> x^2 – 2hx + h^2 + y^2 – 2ky + k^2 = r^2

• Can simplify: x^2 + y^2 – 2hx – 2ky + (h^2+k^2 – r^2) = 0

• Useful for identifying circle from general quadratic equation

Example: Standard: (x−2)² + (y+3)² = 16
Expanded: x² − 4x + 4 + y² + 6y + 9 = 16 → x² + y² − 4x + 6y − 3 = 0