\section*{Problem 4 - 081034}

A quadratic function of the form \(y=x^{2}+p x+q\) is represented in a Cartesian coordinate system.

The points of intersection of the graph of the function with the x-axis bound a segment on it with a length of 7 units. The graph of the function intersects the y-axis at the point \(S_{y}(0 ; 8)\).

Determine the real numbers \(p\) and \(q\)!
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Given the point \(S_{y}\) on the parabola, \(q=8\). The zeros of the function can be expressed as \(-\frac{p}{2} \pm \sqrt{\frac{p^{2}}{4}-q}\), so their distance is calculated as \(2 \sqrt{\frac{p^{2}}{4}-q}=\sqrt{p^{2}-4 q}\). According to the problem, this distance is equal to 7, so \(p^{2}-4 q=49\) or \(p^{2}=81\), which results in \(p= \pm 9\).

Thus, two solution pairs are obtained: \((p, q)=(-9,8)\) or \((p, q)=(9,8)\). The verification confirms both results.