# Find values of variable for which trigonometric expression takes only +ve values?

For what value of $a$ in the interval $\left[\frac{7\pi}{6},\frac{7\pi}{4}\right]$ does the quadratic expression $x^2 \cot a + 2x\sqrt{\tan a} + \tan a$, assume only positive values.
First of all, $\tan a\geq 0\Rightarrow x\in\left[\frac{7\pi}{6},\frac{5\pi}{4}\right]$
$\Delta=b^2-4ac=4\tan a-4<0\Rightarrow\tan a<1\Rightarrow x\in\left[\frac{7\pi}{6},\frac{5\pi}{4}\right)$