More quadratic equations

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.

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One Response to More quadratic equations

  1. Pingback: Don’t disturb my circles | New-cleckit dominie

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