The difference between the corresponding roots of the equations $\displaystyle x^{2} + ax + b = 0$ and $\displaystyle x^{2} + bx + a = 0$ is same, then

Let the difference between the roots of x^2 + ax + b = 0 be d, so sqrt(a^2 - 4b) = d. The difference between the roots of x^2 + bx + a = 0 is also d, meaning sqrt(b^2 - 4a) = d. Equating the two expressions gives a^2 - 4b = b^2 - 4a. Factoring this as a difference of squares, (a - b)(a + b + 4) = 0. Since a cannot equal b for the difference to be uniquely defined and positive, the sum must be zero, yielding a + b + 4 = 0.

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