Algebra 2 · Unit 8 · Topic 8.2
Circles, Centers, and Completing the Square
A circle with center (h, k) and radius r > 0 has equation (x − h)² + (y − k)² = r².
Definition and explanation
A circle with center (h, k) and radius r > 0 has equation (x − h)² + (y − k)² = r². Completing the square can reveal these features from an expanded equation.
Worked example
Rewrite x² + y² − 6x + 4y − 12 = 0. Move the constant and complete squares to obtain (x − 3)² + (y + 2)² = 25. The center is (3, −2) and the radius is 5.
Common mistake
The right side equals radius squared, so 25 gives a radius of 5, not 25.
Key takeaway
A circle with center (h, k) and radius r > 0 has equation (x − h)² + (y − k)² = r².
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.