1. What does "completing the square" mean?
"Completing the square" means changing expressions such as x2 + 4x into the "nearest" perfect square.
In this case if 4 is added then x2 + 4x becomes x2 + 4x + 4 which is the perfect square (x + 2)2.
2. Why do I need to be able to "complete the square" when dealing with the equation of a circle?
If a circle is given in an expanded form such as x2 + y2 + 4x − 6y − 16 = 0 then in order to write it in the form
(x − a)2 + (y − b)2 = r2 the parts x2 + 4x and y2 − 6y must be made into perfect squaresBy adding 4, x2 + 4x becomes x2 + 4x + 4 which equals (x + 2)2
By adding 9, y2 − 6y becomes y2 − 6y + 9 which becomes (x − 3)2
Therefore, x2 + y2 + 4x − 6y = 16
Becomes (x + 4)2 + (x − 3)2 =16 + 4 + 9
This is a circle of radius √29 at centre (-4, 3)
