SAT Circle Equations: Completing the Square Step-by-Step
Master standard circle form and completing the square.
The standard circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. When given an expanded circle equation like x² + y² + 6x - 8y = 24: 1. Group x and y terms: (x² + 6x) + (y² - 8y) = 24. 2. Complete square: add (b/2)² to both sides. 3. Factor: (x + 3)² + (y - 4)² = 49. 4. Center is (-3, 4) and radius r = √49 = 7.
Circle equation questions appear on almost every SAT Math exam. You can also type expanded circle equations directly into Desmos to read the center and radius.
Step-by-Step Breakdown & Rules
1. Group x and y Terms Together
Rearrange equation: (x² + Ax) + (y² + By) = constant.
2. Calculate (b/2)² for Both Variables
Divide the coefficient of x by 2 and square it: (A/2)². Do the same for y: (B/2)².
3. Add the Values to BOTH Sides of Equation
(x² + Ax + (A/2)²) + (y² + By + (B/2)²) = constant + (A/2)² + (B/2)².
4. Factor into Standard Form
(x - h)² + (y - k)² = r² and take the square root of the right side to find radius r.
Key Takeaways & Pro Tips
Frequently asked questions
Is the circle equation formula given on the SAT reference sheet?
No. The circle equation (x-h)² + (y-k)² = r² is NOT provided on the reference sheet and must be memorized.
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