Which of the following quadratic equations has complex roots?

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Question: Which of the following quadratic equations has complex roots?

Options:

Correct Answer: x^2 + 4x + 5 = 0

Solution:

The discriminant of x^2 + 4x + 5 is negative (16 - 20), indicating complex roots.

Which of the following quadratic equations has complex roots?

Which of the following quadratic equations has complex roots?

Step 1: Identify the quadratic equation you want to analyze. In this case, it is x^2 + 4x + 5.
Step 2: Recall the formula for the discriminant of a quadratic equation ax^2 + bx + c, which is given by D = b^2 - 4ac.
Step 3: Identify the values of a, b, and c from the equation. Here, a = 1, b = 4, and c = 5.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = (4)^2 - 4(1)(5).
Step 5: Calculate (4)^2, which is 16.
Step 6: Calculate 4(1)(5), which is 20.
Step 7: Now, subtract the two results: D = 16 - 20.
Step 8: Calculate the result of the subtraction: D = -4.
Step 9: Determine the nature of the roots based on the value of the discriminant. If D is negative, the roots are complex.
Step 10: Since D = -4 is negative, conclude that the quadratic equation x^2 + 4x + 5 has complex roots.

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