First order approximation

How can I neglect higher orders of approximation in Mathematica?

Suppose I want to find the roots of a simple quadratic equation like

x^2 + (b + Epsilon)*x + c == 0


where Epsilon << 1. Obviously, the roots are

x -> 1/2 (-b - Epsilon - Sqrt[-4 c + (b + Epsilon)^2])
x -> 1/2 (-b - Epsilon + Sqrt[-4 c + (b + Epsilon)^2])


Now, how can I neglect Epsilon^2?

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You can use Series to specify the order of approximation. When an expression involving the output of Series, which is a SeriesData object, is evaluated, the calculus is done for you.
sol = Solve[x^2 + (b + Epsilon)*x + c == 0, x]

Alternatively, you could apply Series to the formula directly (use Normal to chop off the big-Oh term).
Series[x /. sol, {Epsilon, 0, 1}]