# Completing the Square/Finding Center & Radius of Circle

• TrueStar
In summary, when solving equations, it is often helpful to work in standard form. This allows for easy graphing of the equation.
TrueStar

## Homework Statement

Find the center and radius of the circle with the given equation:

x^2+y^2+4y-117=0

## Homework Equations

Completing the Square

Formula of a Circle
(x-h)^2-(y-k)^2=r^2

## The Attempt at a Solution

All of the problems I've encountered like this involve completing the square. I haven't seen one where there is no X and coefficient (the same goes for Y). I was able to correctly calculate that the y-coordinate of this circle is -2. The x-coordinate is supposed to be 0 and the radius is 11, but I don't know the steps to get there. I have no x coefficient to complete the square with.

Thank you!

TrueStar said:
x^2+y^2+4y-117=0

All of the problems I've encountered like this involve completing the square. I haven't seen one where there is no X and coefficient (the same goes for Y). I was able to correctly calculate that the y-coordinate of this circle is -2. The x-coordinate is supposed to be 0 and the radius is 11, but I don't know the steps to get there. I have no x coefficient to complete the square with.

Not having an x term actually makes it easier. After all, x^2 = (x - 0)^2, right?

x^2+y^2+4y-117=0, also commonly notated as
x2 + y2 + 4y - 117 = 0, may be rearranged to
x2 + y(y + 4) - 117 =0,
x2 + y(y + 4) = 117

The expression with 'y' can represent a rectangle with "length" y+4 and "height" of y. This rectangle can be itself rearranged to show a missing square section. The process of completing the square is the arithmetic addition of this missing square piece. When you add this square piece to complete the square, you must do so to both sides of the equation. The goal is to be able to set the equation into standard form; it was originally given in general form. Standard form permits you to easily graph the circle.

Yes, this makes sense. I still don't understand how the radius equals to 11 though..unless they rounded the square root of 113.

No, they didn't round the square root of 113. How did you get 113?

You have x^2 + y^2 + 4y - 117=0
so x^2 + y^2 + 4y = 117

What do you have to add to the left side to complete the square? You need to add the same amount to the right side.

I'm sorry, I subtracted instead of added 4. Adding 4 to 117 is 121 and the square root of 121 is 11.

Thanks for explaining this everyone. It was much simpler than I made it out to be.

Find the standard form of the equation 16x^2+9y^2+64x-18y-71= 0

You should start a new thread rather than highjack an existing thread.

## 1. What is Completing the Square and when is it used?

Completing the Square is a method used to solve quadratic equations by rewriting them in a standard form. This method is commonly used when solving for the roots of a quadratic equation or when finding the center and radius of a circle.

## 2. How do you complete the square?

To complete the square, follow these steps:
1. Write the quadratic equation in standard form: ax² + bx + c = 0
2. Divide both sides by the coefficient of x² (a)
3. Take half of the coefficient of x (b) and square it (b/2)²
4. Add the squared value to both sides of the equation
5. Factor the perfect square trinomial on the left side
6. Take the square root of both sides
7. Add or subtract the constant term on the left side, depending on the sign of b
8. Simplify and solve for x

## 3. How do you find the center and radius of a circle using completing the square?

To find the center and radius of a circle using completing the square, follow these steps:
1. Rewrite the equation of the circle in standard form: (x-h)² + (y-k)² = r²
2. Use completing the square on both the x and y terms to put the equation in standard form
3. The values of h and k will be the coordinates of the center of the circle
4. The square root of r² will give you the radius of the circle.

## 4. Can completing the square be used on any type of quadratic equation?

Yes, completing the square can be used to solve any quadratic equation, as long as the coefficient of x² is not equal to 0.

## 5. Are there any alternative methods for finding the center and radius of a circle?

Yes, there are other methods for finding the center and radius of a circle, such as using the distance formula or using the equation of a circle in general form. However, completing the square is a commonly used and efficient method for finding the center and radius of a circle.

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