Completing the Square for Circles

Algebra 2 · Axiom Academy

LESSON Completing the Square for Circles Turn a circle's general-form equation into standard form by completing the square — and read its center and radius straight off the result. 1. The Core Move: Completing a Square To turn x^2 + bx into a perfect square, take half of b , square it, and add that one term. Picture it as area: the x^2 square plus the bx rectangle wraps around two sides and is missing exactly one small corner — and is the piece that fills it. Add the square of half the coefficient A circle needs the square completed twice — once in x , once in y . Whatever you add on the left to complete a square, you must add on the right to keep the equation true. Start from x^2 + y^2 - 6x + 8y = 0 . Add 9 for x and 16 for y — to both sides The right side collects them: 9 + 16 = 25 Standard form is a map. (x-h)^2 + (y-k)^2 = r^2 is the circle centered at (h,k) with radius r . So (x-3)^2 + (y+4)^2 = 25 is centered at (3,-4) with radius . You converted a circle from general form to standard form by completing the square — and read its center and radius straight off the result. Scroll up to revisit any step.

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