Here’s a further undergraduate contribution to the theory of quadratic equations, taken from my most recent plague of marking. The context is a final-year exam; notation has been changed to protect the guilty, but not enough to affect the difficulty of the task.

The task in question: given the equation

$\Omega^2 m + 2\gamma\Omega - \delta m = 0$,

where $\gamma$ and $\delta$ are constants, solve for $\Omega(m)$, then find $\mathrm{d}\Omega/\mathrm{d}m$.

Out of twenty students, three arrived independently at the solution

$\Omega = \pm\sqrt{ \delta -2\dfrac{\gamma\Omega}{m} }$,

and differentiated happily on the basis that $\Omega$ on the RHS was a constant. One arrived at the solution

$\Omega = \dfrac{\delta m}{\Omega m + 2\gamma}$,

and likewise differentiated assuming that $\Omega$ on the RHS was a constant. And one student wrote:

As I can’t reduce only one side to “$\Omega$”, I am assuming $\Omega = \pm\left(\delta -\dfrac{2\gamma}{m}\right)^{1/2}$.

And one examiner wept silently into his coffee.