The analytic solution of the equation $$ 1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{\Lambda}{3}r^2 = 0 $$ obtained by using the code
Solve[1 + Q^2/r^2 - (2 M)/r - (r^2 Λ)/3 == 0, {r}]
is lengthy and complicated.
How can one verify that two approximate solutions obtained by retaining terms only up to first order in $\Lambda$ and $Q^2$, which are small quantities, are $$ r_1 = 2M - \frac{Q^2}{2M} + \frac{4M}{3}(2M^2-Q^2)\Lambda, \\ r_2 = \sqrt{\frac{3}{\Lambda}} -M - \frac{\sqrt{3}}{6}(3M^2-Q^2)\sqrt{\Lambda} $$
Series[r /. Solve[1 + Q^2/r^2 - (2 M)/r - (r^2 \[CapitalLambda])/3 == 0, {r}][[4]], {Q, 0, 2}] // Normal // Series[#, {\[CapitalLambda], 0, 1}] & // FullSimplify[#, Assumptions -> {M > 0}] &
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