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Determine the set of all real values of 'a' for which exactly two roots of the equation (a - 1)(x⁴ + x² + 1) + (a + 1)(x² + x + 1)² = 0 are real and distinct.

1. (0, 1/2)
2. (-1/2, 0) union (0, 1/2)
3. (-1/2, 0)
4. (-inf, -2) union (2, inf)

Correct answer: (-1/2, 0) union (0, 1/2)

Solution

Factoring gives (x²+x+1)[(a-1)(x²-x+1)+(a+1)(x²+x+1)] = 0. Since x²+x+1 has no real roots, the quadratic a*x²+x+a = 0 must supply the two real distinct roots, requiring 1-4a² > 0 and a != 0, i.e., a in (-1/2,0) union (0,1/2).

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