Sum of equation's roots

Question

Find the sum of all real roots of all polynomials in the form of $y=x^2+px+122$, where $p$ can be all integers in the range $[-35,17]$

I could only do it via excel) Which is not mathematical proof..

Answer 1

Hint. (1) Your polynomial has real roots iff $\frac{p^2}4 \ge 122$ or $|p| \ge \sqrt{488} \approx 22.1$

(2) Recall the formula of Vieta, that says that if $x_1$ and $x_2$ are the roots of $x^2 + px + q$, then $p = -(x_1 +x_2)$ and $x_1x_2 = q$.

(3) Recall the Gauß' sum, $\sum_{k=1}^n k = \frac 12n(n+1)$.