To determine which discriminant corresponds to a quadratic equation with complex roots, recall the discriminant formula:

$\Delta = b^2 - 4ac$

The nature of the roots of a quadratic equation depends on the discriminant ($\Delta$):

1. If $\Delta > 0$, the roots are real and distinct.
2. If $\Delta = 0$, the roots are real and equal.
3. If $\Delta < 0$, the roots are complex conjugates.

We need to evaluate the discriminants for each equation and identify the one with $\Delta < 0$.

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Option A: $(6)^2 - 4(1)(5)$

$\Delta = 36 - 20 = 16 \quad (\Delta > 0, \text{real roots})$

Option B: $(-3)^2 - 4(3)(1)$

$\Delta = 9 - 12 = -3 \quad (\Delta < 0, \text{complex roots})$

Option C: $(7)^2 - 4(-2)(12)$

$\Delta = 49 - (-96) = 49 + 96 = 145 \quad (\Delta > 0, \text{real roots})$

Option D: $(1)^2 - 4(-1)(6)$

$\Delta = 1 - (-24) = 1 + 24 = 25 \quad (\Delta > 0, \text{real roots})$

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Conclusion:

The discriminant that corresponds to a quadratic equation with complex roots is:

B. $(-3)^2 - 4(3)(1)$

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Would you like a more detailed explanation of discriminants and their role in quadratic equations? Here are 5 related questions:

1. What happens if the discriminant equals zero?
2. How do discriminants apply in higher-degree polynomials?
3. Can the discriminant be used to classify the nature of roots for all quadratic equations?
4. How do you derive the discriminant formula from the quadratic formula?
5. What are some real-world applications of using discriminants?

Tip: Always check the sign of the discriminant first when analyzing the roots of a quadratic equation!