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If a quadratic equation with real coefficients has roots that are purely imaginary, what can be said about the nature of the roots of the equation formed by substituting the quadratic into itself?

1. All roots are purely imaginary.
2. All roots are real numbers.
3. There are two real roots and two purely imaginary roots.
4. The roots are neither real nor purely imaginary.

Correct answer: The roots are neither real nor purely imaginary.

Solution

If the roots of the original quadratic are purely imaginary, their substitution into the same quadratic equation results in complex roots that are neither purely real nor purely imaginary. This happens because the substitution introduces additional terms that mix real and imaginary components.

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