$If\quad the\quad equations\quad { x }^{ 2 }+ax+b=0\quad and\quad { x }^{ 2 }+px-q=0\quad have\quad a\quad coomon\quad root\ then\quad that\quad root\quad is$

Let the common root be r. Since r satisfies both equations, we have r^2 + ar + b = 0 and r^2 + pr - q = 0. Subtracting the second equation from the first eliminates r^2, giving (a - p)r + (b + q) = 0. Solving this linear equation for r yields r = (b + q) / (p - a). The common root is (b + q) / (p - a).

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