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### Complete This Square: x^2 + y^2 - 6x + 2 = 0

```Date: 03/16/2003 at 23:40:35
From: Peter
Subject: Completing THIS square: x^2 + y^2 - 6x + 2 = 0

Complete the square for:

x^2 + y^2 - 6x + 2 = 0

It has a y variable, which makes it confusing; plus, I thought
completing the square only involved quadratic functions, such as
f(x) = ax^2 + bx + c = 0.

This question relates to the equation of a circle; that is why I need
to complete the square (to figure out its radius and loci/centre).
What's the difference between a loci and centre again? Maybe you can
answer that too please. :) Thanks. Oh, and if it helps, the centre is
(3,0) and the radius is sqrt(7).
```

```
Date: 03/17/2003 at 01:24:42
From: Doctor Jeremiah
Subject: Re: Completing THIS square: x^2 + y^2 - 6x + 2 = 0

Hi Peter,

When you have multiple variables you treat them as completely
unrelated. In fact you can just ignore everything except the quadratic
you are completing the square for. In this equation, the stuff in
parentheses is the only important part:

(x^2 - 6x + 2) + y^2 = 0

So all we really need to be concerned with is:

(x^2 - 6x + 2)
(x^2 - 6x + 2 + 7) - 7
(x^2 - 6x + 9) - 7
(x-3)^2 - 7

Now we have completed the square for the stuff in parentheses. We can
substitute that back in:

(x^2 - 6x + 2) + y^2 = 0
(x-3)^2 - 7 + y^2 = 0
(x-3)^2 + y^2 = 7

So this equation would have its center when x=3 and y=0 and the square

x^2 + y^2 - 16x + 12y + 19 = 0

Isolate the part you will complete the square for:

(x^2 - 16x + 19) + y^2 + 12y = 0
(x^2 - 16x + 19 + 45) - 45 + y^2 + 12y = 0
(x^2 - 16x + 64) - 45 + y^2 + 12y = 0
(x-8)^2 - 45 + y^2 + 12y = 0

Now do the same for the other square:

(x-8)^2 + (y^2 + 12y - 45) = 0
(x-8)^2 + (y^2 + 12y - 45 + 81) - 81 = 0
(x-8)^2 + (y^2 + 12y + 36) - 81 = 0
(x-8)^2 + (y+6)^2 - 81 = 0

Which gives us:

(x-8)^2 + (y+6)^2 = 81
(x-8)^2 + (y+6)^2 = 9^2

http://www.mathworld.wolfram.com/  says a locus is "The set of all
points (usually forming a curve or surface) satisfying some condition.
For example, the locus of points in the plane equidistant from a given
point is a circle, and the set of points in three-space equidistant
from a given point is a sphere."  And loci is the plural of locus.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Conic Sections/Circles
College Definitions
High School Basic Algebra
High School Conic Sections/Circles
High School Definitions

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