= Solution
Put $x=\alpha^2$ and $p=\pi^2$. Then
$$
R_c=\frac{(x+p)^3+Qp(x+p)}x,\qquad
\frac{dR_c}{dx}=\frac{(x+p)^2(2x-p)-Qp^2}{x^2}.
$$
The minimizing condition is
$$
2x^3+3px^2=p^3+Qp^2.
$$
Its left side increases strictly for $x>0$ and the neutral curve diverges at both ends, so there is a unique global minimum. For large $Q$, $x\to\infty$ and the leading balance is $2x^3\sim Qp^2$. Therefore
$$
\boxed{\alpha\sim(\pi^4/2)^{1/6}Q^{1/6}.}
$$
This is the <strong-field wavenumber selection in magnetoconvection> scaling. In a box of fixed width one compares the two admissible horizontal <wavenumbers> bracketing the continuum optimum.
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