= Dissociating isothermal shock
{title2=$2\chi^2-(\mathcal M_1^2+1)\chi+\mathcal M_1^2=0$}
Suppose molecular <hydrogen> is fully dissociated into atoms across a steady planar <shock>, with the same far-upstream and far-downstream <temperature> and negligible binding-energy mass correction. The upstream <isothermal sound speed> obeys $a_1^2=k_BT/(2m_H)$, while $a_2^2=2a_1^2$. Conservation of mass gives $\chi=\rho_2/\rho_1=u_1/u_2$. Momentum conservation then gives $u_1^2+a_1^2=u_1^2/\chi+2a_1^2\chi$, producing the displayed quadratic with $\mathcal M_1=u_1/a_1$.
The high root approaches $\mathcal M_1^2/2$ at large upstream Mach number. Its real compressive domain is $\mathcal M_1\ge1+\sqrt2$. At the endpoint the downstream flow is sonic; above it the high root is subsonic downstream. Completion of dissociation and maintenance of equal temperatures additionally require a consistent chemical and radiative energy budget. The two flux equations alone do not establish that physical realizability.
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