Completing the Square in Circle Equations: Step-by-Step Guide
To complete the square in circle equations, rewrite the general form x² + y² + Dx + Ey + F = 0 by grouping the x-terms and y-terms, adding (D/2)² and (E/2)² to both sides, and factoring each group into a perfect square. The result is the standard form (x − h)² + (y − k)² = r², which reveals the circle’s center (h, k) and radius r. For example, x² + y² − 6x + 4y − 12 = 0 becomes (x − 3)² + (y + 2)² = 25, a circle centered at (3, −2) with radius 5. The technique works whenever a circle’s equation appears in expanded form and you need its center or radius.
Why Completing the Square
A circle’s equation can be written in two ways, but only one of them tells you anything useful at a glance. The general form x² + y² + Dx + Ey + F = 0 is compact, but the center and radius are buried inside it. The standard form (x − h)² + (y − k)² = r² hands you the center and radius directly.
Completing the square is the bridge between the two. It converts an expanded equation into a sum of perfect squares, which is exactly what the standard form requires. You need it whenever you’re asked to find a circle’s center, radius, graph, or diameter from an equation that isn’t already in standard form. The method follows the same logic as completing the square for quadratic expressions.
Standard vs General Form
The two forms describe the same circle — they just organize the information differently.
| Form | Equation | What you can read off |
|---|---|---|
| Standard | (x − h)² + (y − k)² = r² | Center (h, k), radius r |
| General | x² + y² + Dx + Ey + F = 0 | Nothing directly; requires conversion |
In standard form, the numbers subtracted from x and y are the coordinates of the center. Note the sign flip: (x − 3)² means the center’s x-coordinate is 3, while (y + 2)² means the y-coordinate is −2. The value on the right side is r², not r. If the right side is 25, the radius is 5. Forgetting to take the square root is one of the most common errors.
A key detail: both x² and y² must have a coefficient of 1 before you begin. If the equation starts with 2x² + 2y², divide every term by 2 first. And if the right side works out to zero or a negative number, the equation isn’t a real circle — it’s a single point or has no graph at all.
How to read standard form quickly
- (x − 5)² + (y − 1)² = 16 → center (5, 1), radius 4
- (x + 4)² + (y − 0)² = 9 → center (−4, 0), radius 3
- x² + (y + 7)² = 2 → center (0, −7), radius √2
Step-by-Step Method
The process has four reliable steps. Follow them in order and the algebra stays manageable.
- Rearrange and group. Move the constant to the right side. Group the x-terms together and the y-terms together.
- Complete the square for x. Take half of the x-coefficient, square it, and add it to both sides.
- Complete the square for y. Do the same with the y-coefficient.
- Factor and simplify. Write each group as a perfect square and combine the constants on the right.
Worked example
Convert x² + y² − 8x + 6y − 11 = 0 to standard form.
Step 1 — Group and move the constant: (x² − 8x) + (y² + 6y) = 11
Step 2 — Complete the square for x: Half of −8 is −4; (−4)² = 16. Add 16 to both sides. (x² − 8x + 16) + (y² + 6y) = 11 + 16
Step 3 — Complete the square for y: Half of 6 is 3; 3² = 9. Add 9 to both sides. (x² − 8x + 16) + (y² + 6y + 9) = 11 + 16 + 9
Step 4 — Factor and simplify: (x − 4)² + (y + 3)² = 36
The circle has center (4, −3) and radius 6.
Two habits prevent most mistakes here. First, whatever you add on the left, you must add on the right — both 16 and 9 in this example. Second, keep the sign straight when factoring: x² − 8x + 16 factors as (x − 4)², not (x + 4)². Purplemath’s walkthrough of the technique reinforces this sign discipline.
Practice Examples
Try each problem before reading the solution. They progress from straightforward to slightly trickier.
Example 1: x² + y² + 10x − 4y + 13 = 0
Group: (x² + 10x) + (y² − 4y) = −13 Half of 10 is 5, so add 25. Half of −4 is −2, so add 4. Add both to the right side. (x + 5)² + (y − 2)² = −13 + 25 + 4 = 16 Center (−5, 2), radius 4.
Example 2: x² + y² − 12x = 0
There’s no y-term beyond y², and no constant — so the constant is 0. (x² − 12x) + y² = 0 Half of −12 is −6; add 36 to both sides. (x − 6)² + y² = 36 Center (6, 0), radius 6. This is a circle that passes through the origin, which is why F = 0.
Example 3: 2x² + 2y² + 8x − 12y − 6 = 0
Divide everything by 2 first: x² + y² + 4x − 6y − 3 = 0. Group: (x² + 4x) + (y² − 6y) = 3 Half of 4 is 2; add 4. Half of −6 is −3; add 9. (x + 2)² + (y − 3)² = 3 + 4 + 9 = 16 Center (−2, 3), radius 4.
Example 4 (no real circle): x² + y² − 2x + 4y + 9 = 0
Group: (x² − 2x) + (y² + 4y) = −9 Add 1 and 4 to both sides. (x − 1)² + (y + 2)² = −9 + 1 + 4 = −4
A squared quantity can’t sum to a negative number, so this equation has no real graph. If the right side had been 0, it would represent the single point (1, −2). Recognizing this case is just as important as solving the standard ones. Math is Fun’s circle reference is a helpful companion for checking these edge cases.
Common mistakes to avoid
- Forgetting to add the new constants to the right side as well.
- Taking half the coefficient but forgetting to square it.
- Reporting r² as the radius instead of taking the square root.
- Skipping the division step when x² or y² has a leading coefficient other than 1.
Key Takeaways
- Completing the square converts the general form x² + y² + Dx + Ey + F = 0 into standard form (x − h)² + (y − k)² = r².
- The center is (h, k), read with flipped signs from the squared terms; the radius is the square root of the right side.
- Add (D/2)² and (E/2)² to both sides — balancing the equation is essential.
- Divide by the leading coefficient first if x² or y² isn’t already 1.
- If the final right side is negative, there’s no real circle; if it’s zero, the graph is a single point.
