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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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    – bbgodfrey
    Sep 19 '15 at 18:49
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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]
approx = sol /. Epsilon -> Series[Epsilon, {Epsilon, 0, 1}] // Normal

Mathematica graphics

Alternatively, you could apply Series to the formula directly (use Normal to chop off the big-Oh term).

Series[x /. sol, {Epsilon, 0, 1}]

Mathematica graphics

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