Expanded circle equations exist mainly to make you complete the square under time pressure. Desmos graphs any form and shows you the center by eye.
The move: Type the circle, read center & radius.
Why it works: The calculator draws the circle from any algebraic form, so the center and radius become visible facts — remember the signs inside the parentheses flip.
The pattern
- Type the equation exactly as printed
- See the circle appear
- Read the center (signs flip) and radius
Worked example: What is the center of the circle (x − 3)² + (y + 1)² = 25?
- Type it exactly as printed.
(x-3)^2+(y+1)^2=25 - There's the circle — visibly centered at (3, −1). The signs FLIP from the equation: (x − 3) → 3, (y + 1) → −1.
(3,-1)
Worked example 2: What is the radius of the circle (x − 3)² + (y + 1)² = 25?
- Same circle.
(x-3)^2+(y+1)^2=25 - The 25 is r² — so r = 5. Confirm visually: the circle runs from x = −2 to x = 8, a width of 10 = 2r.
Remember it on test day
Type the circle, flip the signs. That's the whole cue — say it when you see this question type and the steps follow.
When this isn't the move
If the question asks you to rewrite into standard form, the graph verifies your algebra but can't produce the expression.
Drill it until it's reflex
This move is one of Desvantage's 18 training modules: a plain-English lesson, a type-it-yourself walkthrough inside a live Desmos calculator, and a 100-question drill bank that deals ten fresh questions per round. The first two modules — plus the 12-question diagnostic — are free.