$If\quad the\quad equations\quad { x }^{ 2 }+ax+b=0\quad and\quad { x }^{ 2 }+bx+a=0\ have\quad a\quad common\quad root,\quad then$

Let the common root be r. Since r is a root of both x^2 + ax + b = 0 and x^2 + bx + a = 0, subtracting the two equations eliminates r^2. This leaves (a - b)r + (b - a) = 0, which factors to (a - b)(r - 1) = 0. Assuming the equations are distinct and a does not equal b, the common root must be r = 1. Substituting x = 1 back into the first equation gives 1^2 + a(1) + b = 0, which simplifies to a + b = -1.

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