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When \(a \ne 0\), there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$ or \[x = {-b \pm \sqrt{b^2-4ac} \over 2a}.\]
Classic Stefan Problem
Physical problems which include a phase change are frequent in the natural world. This is particularly true in water for which all three phases are attainable at normal atmospheric temperatures. Some examples include a frozen pond that melts in the spring time or molten lava which will solidify when exposed to the atmosphere. Phase change problems are complicated in that they include a moving boundary. A bulk of the original work on phase change was done by Jo\v{z}ef Stefan in the late 19th century, but since then there has been a lot of development on what we call the Stefan problem.\par In order to address the phase change problem we first need to establish the conservation of energy within some domain. The constitutive relation for energy, known as Fourier's law, is \begin{equation} Q = -\kappa \nabla T \end{equation} where \(Q\) is the heat flux, \(\kappa\) is the thermal conductivity of the material, and \(T\) is the temperature. This equation says that energy will move down the temperature gradient by conduction. Conservation of energy by the first law of thermodynamics combined with the above constitutive relation gives us the heat equation \begin{equation}\label{eq:Heat} \frac{\partial T}{\partial t} = \alpha \nabla^2 T \end{equation} where the thermal diffusivity \(\alpha = {\kappa}/\left({\rho C_p}\right)\) describes the rate of heat diffusion for the material properties of density, \(\rho\), and specific heat capacity at constant pressure, \(C_p\).Classic Stefan Problem
A test of Equation References
Here's one using the equation environment: \begin{equation} x+1\over\sqrt{1-x^2} \end{equation} and one with equation* environment: \begin{equation*} x+1\over\sqrt{1-x^2} \end{equation*}
This is a forward reference [\ref{ref2}] and another \eqref{ref2} for the following equation: $$x+1\over\sqrt{1-x^2}\label{ref2}$$ More math: $$x+1\over\sqrt{1-x^2}$$ Here is a ref inside math: \(\ref{ref2}+1\) and text after it. \begin{align} x& = y_1-y_2+y_3-y_5+y_8-\dots && \text{by \eqref{ref1}}\\ & = y'\circ y^* && \text{(by \eqref{ref3})}\\ & = y(0) y' && \text {by Axiom 1.} \end{align} Here's a bad ref [\ref{ref4}] to a nonexistent label.
An alignment: \begin{align} a&=b\label{ref3}\cr &=c+d \end{align} and a starred one: \begin{align*} a&=b\cr &=c+d \end{align*}