ACT® · Math: Geometry — Coordinate Geometry and Conic Sections
Completing the Square for a Circle
Completing the square turns a messy x2 + y2 equation into (x-h)2 + (y-k)2 = r2, the form that shows the circle's center (h, k) and radius r directly.
The equation x2 + y2 - 6x + 8y = 0 is a circle in disguise. It's built from the standard form (x-h)2 + (y-k)2 = r2 but someone expanded the squares and mixed everything together. Expand (x-h)2 and you get x2 - 2hx + h2 — so the number multiplying x is -2h. That means whatever coefficient sits on x right now, half of it (with its sign flipped) is the center's x-coordinate. Same logic for y. Completing the square is just the reverse process: rebuilding the perfect square that got torn apart.
Work on x and y separately. Take the x terms, x2 - 6x, grab half the coefficient (-3), square it (9), and add it: x2 - 6x + 9 = (x-3)2. Do the same for y terms: y2 + 8y + 16 = (y+4)2. Every number you add to the left side has to be added to the right side too, or the equation stops being true. Once both sides balance and both squares are built, you read off the center and radius straight from the form (x-h)2 + (y-k)2 = r2.
This skill sits right next to the distance and midpoint formulas because a circle's equation is really just 'distance from center equals r' written out algebraically. Once a circle is in center-radius form, questions about where it sits, whether it touches an axis, or where it intersects a line all become distance and slope problems you already know how to handle.
Key terms
- Completing the square
- A method that adds a specific constant to a quadratic expression so it becomes a perfect square trinomial.
- Center-radius form
- A circle's equation written as (x-h)2 + (y-k)2 = r2, showing center (h, k) and radius r directly.
- Perfect square trinomial
- An expression like x2 - 6x + 9 that factors into a squared binomial, here (x-3)2.
- Radius
- The fixed distance from a circle's center to any point on the circle.
Watch it explained
- How it worksHow does adding the square of half the x coefficient rebuild a perfect square trinomial?
- How it worksHow does completing the square on x and on y separately drag the hidden center into view?
- Worked exampleFind the center and radius of x2 + y2 - 10x + 4y - 7 = 0.
- Worked exampleFind the radius of x2 + y2 + 6x - 2y + 6 = 0 after completing both squares.
- Common mistakeWhy doesn't the constant on the right side stay put while you complete the square on the left?
Check yourself
End of topic.