SAT Math: Complete the square to rewrite a circle equation in standard form
18+ practice questions in Praczo
The concept, explained
- 1
Standard form of a circle: (x - h)² + (y - k)² = r², with center (h, k) and radius r.
- 2
If the equation is expanded (x² + y² + Dx + Ey + F = 0), group x-terms and y-terms, then complete the square for each group.
- 3
To complete the square on x² + Dx, add (D/2)². You must add the same value to BOTH sides of the equation.
- 4
After completing the square: (x + D/2)² + (y + E/2)² = (D/2)² + (E/2)² - F. Read off center (-D/2, -E/2) and radius √(right side).
- 5
Watch the signs: (x + 4)² corresponds to center x = -4, not x = 4.
- ✗ Adding (D/2)² to only one side of the equation, breaking the equality.
- ✗ Flipping the sign when reading center coordinates from (x - h)² — the center is (h, k), not (-h, -k).
- ✗ Taking r² as r (forgetting the square root at the end).
SAT-style practice
What are the center and radius of the circle x² + y² - 6x + 8y - 11 = 0?
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