College Algebra & Trigonometry, 2018a

262 CHAPTER 5. CONIC SECTIONS - CIRCLE AND PARABOLA
We can see that if we have x 2 +6x, then this would correspond to (x +3)(x +3)
or (x +3) 2 . But there’s a problem: (x +3) 2 is not equal to x 2 +6x. It’s equal to
x 2 +6x +9. We can’t just add a 9 to the x 2 +6x, but we can add a 9 to both sides.
x 2 +6x +9+y 2 +10y = −25+9
Similarly, to complete the square on the y, we see that y 2 +10y corresponds to
(x +5)(x +5)or (x +5) 2 . Here, we would need to add 25 to both sides to create a
perfect square.
x 2 +6x +9+y 2 +10y +25=−25+9+25
x 2 +6x +9+y 2 +10y +25=9
(x +3) 2 +(y +5) 2 =9
So, the center of the circle is (−3, −5) and the radius is 3.
Sometimes it is not so obvious what the values of h and k should be in completing
the square. Consider the equation below:
x 2 +20x + y 2 +30y +15=0
If we look back at the examples for squaring binomials, we can see the pattern
that relates the coefficient of the linear term to the values for h and k.
(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
Notice that the coefficient of the linear term is always double the value of the
numeral in the parentheses and the constant term is always that number squared.