The equations $x^5 + ax + 1 = 0$ and $x^6 + ax^2 + 1= 0$ have a common root. Then $a$ is equal to

Let the common root be r, so r^5 + a*r + 1 = 0 and r^6 + a*r^2 + 1 = 0. Multiplying the first equation by r gives r^6 + a*r^2 + r = 0. Subtracting the second given equation from this result eliminates a and 1, yielding r = 0. Since r = 0 makes 1 = 0 in the original equations, it is invalid, meaning we must multiply the second equation by a factor and subtract to find another common root. Multiplying the first equation by r and the second by 1, then subtracting the first from the second gives r^6 + a*r^2 + 1 - (r^6 + a*r^2 + r) = 0, which simplifies to 1 - r = 0, so r = 1. Substituting r = 1 into the first equation, 1^5 + a*1 + 1 = 0, gives a = -2.

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