In what interval can $\displaystyle p+q$ be if we know that the sum of the squares of the roots of $\displaystyle x^2+px+q=0$ equals to 1 ($\displaystyle x_1^2+x_2^2=1$).
there exists some $\displaystyle t$ such that $\displaystyle x_1=\cos t, \ x_2= \sin t.$ let $\displaystyle x=\cos t + \sin t.$ we have $\displaystyle p+q=-(x_1+x_2)+x_1x_2=\frac{1}{2}(x-1)^2 - 1.$ finally use the fact that $\displaystyle -\sqrt{2} \leq x \leq \sqrt{2}$ to finish the proof.