MathGeometry & TrigonometryMedium frequency
SAT Math: Equation of a Circle: Completing the Square
22+ practice questions in Praczo
What you need to know
The concept, explained
- 1
The standard equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
- 2
If given the general form x² + y² + ax + by = c, group the x terms and y terms to complete the square for each.
- 3
To complete x² + ax, add (a/2)² to BOTH sides of the equation.
- 4
This condenses into (x + a/2)². Repeat for the y terms.
- 5
Always remember to add the same values to the right side of the equation; otherwise, you will get the wrong radius.
Common mistakes
- ✗ Completing the square on the left side but forgetting to add those same constants to the right side.
- ✗ Misidentifying the center signs (e.g., if it takes the form (x + 3)², the h coordinate is -3, not 3).
Try a sample question
SAT-style practice
Find the radius of the circle given by the equation x² + y² - 4x + 6y = 12.
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