If the equation $x ^ { 2 } + c x + a b = 0$ and $x ^ { 2 } + b x + c a = 0$ have a common root, then $a + b + c$ is equal to

Let alpha be the common root of x^2 + cx + ab = 0 and x^2 + bx + ca = 0. Using the condition for a common root, we set the cross-determinants equal: alpha^2 is proportional to (c(b) - a(c)), alpha is proportional to (a(1) - ab(1)), and 1 is proportional to (1(b) - 1(c)). This gives the common root alpha as (ab - a) divided by (b - c). Substituting alpha back into the first equation x^2 + cx + ab = 0 gives the relation a(b - 1)^2 + c(a - c)(b - c) + ab(b - c)^2 = 0. Expanding and simplifying this expression yields a^2(b - c) + b^2(c - a) + c^2(a - b) = 0. Factoring this standard symmetric polynomial gives (a - b)(b - c)(c - a) = 0. Since a, b, and c must be distinct for the equations to be different, this algebraic identity only holds true if the fundamental sum a + b + c is equal to 0.

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