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Let us use exact arithmetics (first), instead of numerics. (Using an inaccurate f below also works, when we only needs roots in ring=CC, which uses a given precision, usually a sufficient one.)

alpha0 =  1/4             #  0.25
alpha1 = -5/2             # -2.5
alpha2 =  69282 / 10000   #  6.9282
alpha3 = -55/10           # -5.5
alpha4 =  1/2             #  0.5
alpha5 = -1/2             # -0.5

x = var('x')
f = alpha1 + alpha2*x + alpha3*x**2 + x**4 + alpha4/(x**2 - alpha0) + alpha5*x**2/(x**2 - alpha0)
g = f.factor().numerator()
print "g = %s" % g

I have only invented g. Then one of the following is the solution:

g.roots( ring=QQbar, multiplicities=False )
g.roots( ring=CC   , multiplicities=False )

For instance:

print "g = %s" % g
print "The roots of g are as follows:"
g.roots( ring=QQbar, multiplicities=False )

Results:

g = 20000*x^6 - 115000*x^4 + 138564*x^3 - 32500*x^2 - 34641*x + 22500
The roots of g are as follows:
[-2.875215523839503?,
 -0.5433030528116862?,
 0.9541487172225938?,
 1.408897521688209?,
 0.5277361688701930? - 0.5071712949223566?*I,
 0.5277361688701930? + 0.5071712949223566?*I]