Converting general form to standard form by completing the square — the final piece connecting circles back to algebra.
The Mnemonic
The Same Circle, Written Expanded
The general form of a circle's equation is x² + y² + Dx + Ey + F = 0 — this is exactly the standard form, (x−h)²+(y−k)²=r², but fully MULTIPLIED OUT (expanded), so the center and radius are no longer visible at a glance.
To find the center and radius from general form, you need to reverse the expansion using a technique called completing the square — converting the equation back into standard form, where h, k, and r become readable again.
Completing the square converts the expanded general form back into the readable standard form
💡 Memory Trick
"Converting general form to standard form by completing the square." General form is standard form's disguise — expanded out, with the center and radius hidden inside the D, E, and F coefficients. Completing the square is exactly how you undo that disguise and get back to (x−h)²+(y−k)²=r², where everything is visible again.
Why It Works
Where D, E, and F Actually Come From
Expand the standard form to see exactly where general form comes from: (x−h)² + (y−k)² = r² becomes x²−2hx+h² + y²−2ky+k² = r², which rearranges to x² + y² + (−2h)x + (−2k)y + (h²+k²−r²) = 0. Matching this to general form shows D = −2h, E = −2k, and F = h²+k²−r².
This means the center coordinates are hidden as h = −D/2 and k = −E/2 — but rather than memorizing these formulas directly, completing the square is the more reliable method, since it works through the algebra explicitly and is less prone to sign errors than plugging into a memorized shortcut formula.
Using It In A Proof
The Completing-the-Square Process
Converting from general form to standard form follows a consistent, repeatable process.
1
Group x-terms and y-terms, move F to the other side
Rearrange x² + y² + Dx + Ey + F = 0 into (x² + Dx) + (y² + Ey) = −F.
2
Complete the square for each variable separately
For x² + Dx, add (D/2)² to both sides. For y² + Ey, add (E/2)² to both sides — remembering to add these same amounts to the right side of the equation too.
3
Factor each side into a perfect square
The x-group becomes (x + D/2)², and the y-group becomes (y + E/2)² — giving the full standard form equation.
4
Read off the center and radius
Center = (−D/2, −E/2). Radius = the square root of whatever remains on the right side.
Full Worked Example
Converting a General Form Equation to Standard Form
Given: x² + y² − 6x + 4y − 12 = 0. Find: the center and radius by completing the square.
1
Group and move the constant
(x² − 6x) + (y² + 4y) = 12.
2
Complete the square for x
Half of −6 is −3, and (−3)² = 9. Add 9 to both sides: (x² − 6x + 9) + (y² + 4y) = 12 + 9.
3
Complete the square for y
Half of 4 is 2, and 2² = 4. Add 4 to both sides: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4.
This is the exact same equation-of-a-circle skill from Coordinate Geometry — general form just required this extra completing-the-square step first to reveal it.
🎯 Quick Worked Example
Convert x² + y² + 8x − 2y + 8 = 0 to standard form and identify the center and radius.
1
Group and isolate. (x² + 8x) + (y² − 2y) = −8.
2
Complete both squares. Half of 8 is 4, 4²=16. Half of −2 is −1, (−1)²=1. Add both: (x²+8x+16) + (y²−2y+1) = −8+16+1.
3
Factor and conclude. (x+4)² + (y−1)² = 9. Center = (−4, 1). Radius = √9 = 3.
📌 Exam Application
Watch for a general form equation where F makes the final right-side value NEGATIVE after completing the square — this would mean r² is negative, which is impossible, indicating the equation doesn't actually describe a real circle at all (sometimes called an 'imaginary' or empty circle) — always check that the final constant is positive before taking its square root.
⚠️ Most Common Circle Equation — General Form Mistakes
Trap 1 — Forgetting to add the completing-the-square constants to BOTH sides: Whatever value you add to complete a square on the left must also be added to the right side of the equation — adding it only on one side breaks the equation's balance and gives a wrong radius.
Trap 2 — Sign errors when reading off the center: If the completed square is (x+4)², the center's x-coordinate is −4 (since this is really (x−(−4))²) — the same sign-flip trap that first appeared back in the standard form Equation of a Circle lesson.
✓ Quick Self-Test
1) What is the general form of a circle's equation? 2) What algebraic technique converts general form into standard form? 3) Convert x² + y² − 4x − 6y − 3 = 0 to standard form. 4) Using that same equation, state the center and radius. 5) What does it mean if completing the square leaves a negative value on the right side of the equation?