Question

Find the center and radius for the circle defined by the equation: x2 + y2 + 5x − y + 2 = 0

Answer 1

Answers:
Center = (-2.5, 0.5)
Radius = 2.1213 units approximately

Note: The center written in fraction form is (-5/2, 1/2)

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Work Shown:

x^2 + y^2 + 5x - y + 2 = 0 
x^2 + 5x + y^2 - y + 2 = 0 
x^2 + 5x + y^2 - y = -2
x^2 + 5x + 6.25 + y^2 - y = -2 + 6.25 <<--- see note1 below
(x^2 + 5x + 6.25) + y^2 - y = 4.25
(x + 2.5)^2 + y^2 - y = 4.25
(x + 2.5)^2 + y^2 - y + 0.25 = 4.25 + 0.25 <<--- see note2 below
(x + 2.5)^2 + (y^2 - y + 0.25) = 4.5
(x + 2.5)^2 + (y - 0.5)^2 = 4.5

The equation is in the form (x-h)^2 + (y-k)^2 = r^2
where,
h = -2.5 = -5/2
k = 0.5 = 1/2
r = sqrt(4.5) = 2.1213 approximately

So that's why the center is (-5/2, 1/2) = (-2.5, 0.5) and the radius is approximately 2.1213 units

note1: I took half of 5 to get 5/2 = 2.5 then I squared it to get (2.5)^2 = 6.25; The value 6.25 is added to both sides. This is done to complete the square for the x terms

note2: Similar to note1, but done for the y terms now. I took half of -1 to get -1/2 = -0.5 and then squared it to get (-0.5)^2 = 0.25, which is added to both sides