SAT® · Math: Geometry and Trigonometry
Completing the Square to Find Center and Radius
A circle's equation sometimes shows up expanded and scrambled, with x2 and y2 terms mixed in with lower-degree terms. Completing the square on the x's and y's separately rebuilds it into center-radius form, (x-h)2 + (y-k)2 = r2, where the center and radius are easy to read off.
A circle's equation in standard form, (x-h)2 + (y-k)2 = r2, tells you everything at a glance: center (h, k), radius r. But if you expand those squares, the clean form disappears into something like x2 + y2 - 6x + 8y = 0, with no center or radius visible. Completing the square undoes that expansion. Group the x terms, group the y terms, and for each group take half the coefficient of the linear term, square it, and add it to both sides. That extra piece turns x2 - 6x into (x-3)2 - 9, a perfect square trinomial minus a constant. Do the same for y, move constants to the right side, and the equation snaps back into recognizable form.
Work through x2 + y2 - 6x + 8y = 0. Half of -6 is -3, squared is 9. Half of 8 is 4, squared is 16. Add both to each side: (x2 - 6x + 9) + (y2 + 8y + 16) = 25, which factors to (x-3)2 + (y+4)2 = 25. Center (3, -4), radius 5. If the squared terms carry a coefficient other than 1, like 2x2 + 2y2 - 12x + 20y - 6 = 0, divide every term by that coefficient first, since standard form needs coefficient 1 on both squares. That gives x2 + y2 - 6x + 10y - 3 = 0, then proceed as before.
This is the same move as completing the square to turn a quadratic from standard form into vertex form, a(x-h)2 + k. In both cases you're forcing a messy expression into a shape that exposes a center-like point directly. On the circles side, it connects to the next skill: once you can read off center and radius, you can test whether a point lies inside, on, or outside the circle, or find where a line intersects it.
Key terms
- Standard form of a circle
- The equation (x-h)2 + (y-k)2 = r2, showing center (h, k) and radius r directly.
- Completing the square
- Adding the square of half the linear coefficient to turn an expression like x2 + bx into a perfect square trinomial.
- Center-radius form
- Another name for standard form of a circle, naming what the two key pieces represent.
- Perfect square trinomial
- An expression like x2 - 6x + 9 that factors into a squared binomial, here (x-3)2.
Watch it explained
- How it worksHow does adding the square of half the x coefficient rebuild a perfect square out of x2 + 6x?
- Worked exampleFind the center and radius of x2 + y2 - 6x + 8y = 0 by completing the square twice.
- Worked exampleFind the radius of 2x2 + 2y2 - 12x + 20y - 6 = 0 after dividing through and completing the square.
- Common mistakeWhy doesn't adding 9 to just one side of a circle's equation leave the circle where it was?
- Ties togetherHow does this move match the one that rewrites a quadratic function in vertex form?
Check yourself
End of topic.