Completing the square to find a circle center

Q: How do you find the center and radius from a circle equation in general form?

Group the x-terms and y-terms, complete the square for each group, and rewrite the equation in (x-h)^2 + (y-k)^2 = r^2 form. Then read the center and radius directly.

Q: What does completing the square tell you about a circle equation?

Completing the square turns the general equation into standard form, which reveals the circle’s center and the squared radius. That makes the geometry of the equation easy to see.

Question

2. Find the center and radius of the circle represented by the equation below.

$x^2 + y^2 + 10x + 8y - 40 = 0$

$x^2 + 10x + y^2 + 8y = 40$

$(x-5)^2 + (y+4)^2 = 81$

(center (5, -4)   r = 9

Acemy AI · Accepted Answer

> **Key concept**: This is a standard completing-the-square problem. Rewrite the circle equation into center-radius form, then read off the center and radius directly.

### Step by step
## Key idea: rewrite in standard circle form
The equation of a circle should look like

$$
(x-h)^2 + (y-k)^2 = r^2
$$

To find the center and radius, we complete the square for both the x-terms and y-terms in

$$
x^2 + y^2 + 10x + 8y - 40 = 0
$$

## Step 1: move the constant
Add 40 to both sides:

$$
x^2 + 10x + y^2 + 8y = 40
$$

Now group the x-terms and y-terms:

$$
(x^2 + 10x) + (y^2 + 8y) = 40
$$

## Step 2: complete the square
Take half of 10, square it, and add 25. Take half of 8, square it, and add 16.

$$
(x^2 + 10x + 25) + (y^2 + 8y + 16) = 40 + 25 + 16
$$

$$
(x+5)^2 + (y+4)^2 = 81
$$

## Step 3: identify center and radius
From $ (x+5)^2 + (y+4)^2 = 81 $:
- center is $(-5,-4)$
- radius is $\sqrt{81}=9$

## Important check
The correct center comes from the signs inside the parentheses: $x+5$ means the x-coordinate is $-5$, and $y+4$ means the y-coordinate is $-4$. The sign is reversed when you read the center.

## Final answer
Center: $$
\boxed{(-5,-4)}
$$
Radius: $$
\boxed{9}
$$

### Pitfall alert
A very common mistake is to change the signs incorrectly when identifying the center. If the equation is $ (x+5)^2 + (y+4)^2 = 81 $, the center is $(-5,-4)$, not $(5,4)$. Another error is completing the square on one variable but forgetting to add the same amount to the right side. That breaks the equality and gives an incorrect radius. When checking your work, always expand the final form to see if it matches the original equation.

### Try different conditions
If the equation were $x^2 + y^2 - 10x + 8y - 40 = 0$, the only change would be the sign of the x-term. Completing the square would give $ (x-5)^2 + (y+4)^2 = 81 $, so the center would be $ (5,-4) $ and the radius would still be 9. If the constant term were different, the radius would change as well because the right-hand side after completing the square determines $r^2$.

### Further reading
completing the square, standard form of a circle, circle center and radius

Question

Completing the square to find a circle center

Expert Verified Solution

Step by step

Key idea: rewrite in standard circle form

FAQ

How do you find the center and radius from a circle equation in general form?

What does completing the square tell you about a circle equation?