Question

Find the center and radius for the circle defined by the equation x^2 + y^2 + 5x -y + 2

Answer 1

Answers:

Center is (-2.5, 0.5)
Radius is 2.12 units

Note: the value of the radius is approximate and rounded to 2 decimal places
If you need the exact value in terms of a square root, then the exact radius is sqrt(4.5) units long where 'sqrt' stands for 'square root'

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Work Shown:

We need to complete the square for the x and y terms

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

So the center is (-2.5, 0.5) and the radius is roughly 2.12 units, which is the result of taking the square root of 4.5

sqrt(4.5) = 2.12132 approximately and that rounds further to 2.12

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Note1: I took half of the x coefficient 5 to get 2.5, then I squared that to get 6.25. This value is added and subtracted on the same side to keep things balanced. The goal of doing all this to complete the square for the x terms

Note2: Similar to note1 above, but this time it's for the y terms. In this case, the y coefficient is -1. Half of that is -0.5 which squares to 0.25. The value of 0.25 is added and subtracted on the left side to keep things balanced.