What is the value of k for which the equation x^2 + 2kx + 16 = 0 has real roots?

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Question: What is the value of k for which the equation x^2 + 2kx + 16 = 0 has real roots? (2021)

Options:

Correct Answer: 8

Exam Year: 2021

Solution:

The discriminant must be non-negative: (2k)^2 - 4(1)(16) ≥ 0. This gives 4k^2 - 64 ≥ 0, leading to k^2 ≥ 16, so k ≥ 4.

What is the value of k for which the equation x^2 + 2kx + 16 = 0 has real roots? (2021)

What is the value of k for which the equation x^2 + 2kx + 16 = 0 has real roots? (2021)

Step 1: Identify the equation given: x^2 + 2kx + 16 = 0.
Step 2: Recognize that for a quadratic equation to have real roots, the discriminant must be non-negative.
Step 3: Write down the formula for the discriminant: D = b^2 - 4ac, where a = 1, b = 2k, and c = 16.
Step 4: Substitute the values into the discriminant formula: D = (2k)^2 - 4(1)(16).
Step 5: Simplify the expression: D = 4k^2 - 64.
Step 6: Set the discriminant greater than or equal to zero for real roots: 4k^2 - 64 ≥ 0.
Step 7: Solve the inequality: 4k^2 ≥ 64.
Step 8: Divide both sides by 4: k^2 ≥ 16.
Step 9: Take the square root of both sides: k ≥ 4 or k ≤ -4.
Step 10: Conclude that the values of k for which the equation has real roots are k ≥ 4 or k ≤ -4.

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