In the textbook titled Mathematical Methods for Physics and Engineering by Riley, Hobson and Bence, 3rd edition, there is a section on page 664 on the derivation of the uncertainty principle.
It starts of by considering two state vectors $ \left|u \right>$ and $ \left|v\right>$, and a real scalar $\lambda$. Letting $ \left|w\right> = \left|u \right> + \lambda \left|v \right> $, we take the inner product of $ \left|w\right> $ with itself and remember that the norm squared is always greater or equal to $0$.
$$ 0 \leq \left<w|w\right> = \left<u|u\right> + \lambda(\left<u|v\right> + \left<v|u\right>) + \lambda^2 \left<v|v\right> $$
It then says that this is a quadratic inequality in $\lambda$ and therefore the quadratic equation formed by equating the RHS to zero must have no real roots. What I don't understand is why mustn't it have real roots? We defined that $\lambda$ was real in the first place!