2782.11 – Minding Your p's and q's
Given that , show that . Can a stronger statement be made?
Since , we can work exclusively with . We want to know if the quantity is less than . In other words, we want to solve the inequality,
Our task is now to prove that is always positive. Well, is a quadratic function. Its graph is a parabola. In this case the parabola opens upwards. Such a parabola is always positive if it is never zero. Thus we ask for the solution of the equation:
According to the quadratic formula the solution is:
But since there is no (real) square root of , it follows that is never zero. This proves as desired.
By trying some values, you might guess that in fact, Applying the quadratic formula as we just did, you can show that this is true, except if , when . Therefore,
If you have ever been told to "mind your p's and q's" you may wonder where the expression comes from. The OED has a great article about it--you can google it.