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For what values of a is x an imaginary number in the equation x = a^2 − a −2?

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Final answer:

There are no values of a for which x is an imaginary number in the given equation.

Step-by-step explanation:

To find the values of a for which x is an imaginary number in the equation x = a^2 - a - 2, we need to determine the discriminant of the quadratic equation. The discriminant is given by b^2 - 4ac. If the discriminant is less than 0, then the equation has imaginary solutions.

In our equation, we have a = 1, b = -1, and c = -2. Substituting these values, we have discriminant = (-1)^2 - 4(1)(-2).

Simplifying, we get discriminant = 1 + 8 = 9. Since the discriminant is positive, the equation has no imaginary solutions. Therefore, there are no values of a for which x is an imaginary number in the given equation.

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