College Algebra & Trigonometry, 2018a

5.1. THE EQUATION OF THE CIRCLE 261
Sometimes the equation for a circle is not given in the standard form. In this
situation, you need to put the equation into standard form and then determine
the center and radius. In order to put the equation into standard form you will
need to complete the square. Completing the square is a mathematical technique
that is often useful and is the basis for how the quadratic formula is derived.
Suppose that we are given the equation of a circle that is not in standard form:
x 2 +6x + y 2 +10y +25=0
We need to restate this relationship so that the center and radius can be easily determined
from the equation. In order to do this, we need to complete the square
for both the x and the y variables. There are a variety of methods for completing
the square I will demonstrate one of these below.
Take the original equation and move any term that doesn’t have a variable to the
other side:
x 2 +6x + y 2 +10y = −25
Then open a space after the 6x and the 10y:
x 2 +6x + y 2 +10y = −25
The idea is that we want trinomial expressions for both x and y that are perfect
squares.
If we look at squaring binomial expressions, we can see that there is a pattern:
(x +3) 2 =(x +3)(x +3)=x 2 +6x +9
(x +4) 2 =(x +4)(x +4)=x 2 +8x +16
(x +5) 2 =(x +5)(x +5)=x 2 +10x +25
(x +6) 2 =(x +6)(x +6)=x 2 +12x +36