General Equation of a Circle

The general equation of a circle: complete the square first

Direct answer: For real coefficients, x²+y²+Dx+Ey+F=0 becomes (x+D/2)²+(y+E/2)²=(D²+E²−4F)/4. If the right-hand side is positive, the center is (−D/2,−E/2) and the radius is its positive square root. A zero right-hand side gives one point; a negative value gives no real points.

Four steps to the center and radius

1. Check that x² and y² have the same nonzero coefficient and that there is no xy term. If the common coefficient is A, divide the entire equation by A before identifying D, E and F.
2. Group x²+Dx and y²+Ey. Complete each square by adding (D/2)² and (E/2)², making the same additions on the other side.
3. Write (x−h)²+(y−k)²=K. A circle with radius √K exists only when K>0. Treat K=0 and K<0 separately.
4. Read the center as (h,k), checking the signs inside the brackets. Expand your result back out or substitute a point on the circle to check it.

Worked example: 2x²+2y²+16x−12y=0

Divide every term by 2 to get x²+y²+8x−6y=0. Completing the square gives (x+4)²−16+(y−3)²−9=0, so (x+4)²+(y−3)²=25. The center is (−4,3) and the radius is 5. The point (1,3) is 5 units from the center and satisfies the original equation. Now vary only the constant F in the normalized general form.

Seeing x²+y² is not enough to establish that the equation represents a circle. Normalize the coefficients and check that the completed-square right-hand side is strictly positive.

Common mistakes and limiting cases

• Reading the center's x-coordinate as 4 from (x+4)². Compare with (x−h)²: h=−4.
• Dividing only the squared terms by their common coefficient. The linear and constant terms must be divided too.
• Calling 25 the radius. It is the radius squared; the radius is the positive value 5.
• Treating K=0 as a circle with positive radius, or taking a real square root when K<0. The first case is a single point; the second has no real solution.
• Applying this formula directly to 2x²+y²−1=0. Its squared terms have different coefficients, so the curve needs a separate analysis.

Method reference and example notes

• OpenStax Intermediate Algebra 2e: distance formulas and circles

Last reviewed: 2026-09-11. The reference explains completing the square. The numerical example and parameter comparison are AICSCA practice material, not official exam questions.