Step-by-step explanation:
so, we have
x² + 4x + y² - 6y = -4
we are told that this is actually an equation for a circle and should transform it into the standard form for a circle equation :
(x - a)² + (y - b)² = r²
with (a, b) being the center of the circle, r is the radius.
and yes, the hint is to complete the square(s).
as you can see, in the standard form we have 2 squares besides the constant r² term.
so, let's start with (x - a)².
if this gets fully multiplied. we get
x² - 2ax + a²
which of these terms do we have in our original equation ? we are looking for terms in x ...
we see
x² and 4x.
aha !
this looks like (x + 2)². but it would be
x² + 4x + 4
hmmm. so, what is missing ? the "+ 4" part.
so, we add 4 (remember, on both sides of the equation to keep the original balance of the equation) :
x² + 4x + y² - 6y + 4 = -4 + 4
this gives us
(x + 2)² + y² - 6y = 0
the first square is completed.
note we do the same thing for the y square. we need
(y - b)² = y² - 2by + b²
what terms in y do we have in the original equation ?
y² and -6y
aha, again !
comparing -2by with -6y this suggests b = 3.
so it looks like (y - 3)², but it would be
y² - 6y + 9
what is missing here in our original equation ? the "+ 9" part.
so, we add 9 (again on both sides) :
(x + 2)² + y² - 6y + 9 = 0 + 9
that gives us
(x + 2)² + (y - 3)² = 9
and ta-daaahhhh !
that is it ! we completed both squares.
(-2, 3) is the center of the circle, and its radius is sqrt(9) = 3.
