What is the value of k if the roots of the equation x^2 - kx + 16 = 0 are real a

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Question: What is the value of k if the roots of the equation x^2 - kx + 16 = 0 are real and distinct?

Options:

Correct Answer: 12

Solution:

For the roots to be real and distinct, the discriminant must be positive: k^2 - 64 > 0, leading to k > 8 or k < -8.

What is the value of k if the roots of the equation x^2 - kx + 16 = 0 are real and distinct?

What is the value of k if the roots of the equation x^2 - kx + 16 = 0 are real and distinct?

Step 1: Identify the equation given, which is x^2 - kx + 16 = 0.
Step 2: Understand that for a quadratic equation to have real and distinct roots, the discriminant must be greater than zero.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is calculated as D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = -k, and c = 16. So, the discriminant is D = (-k)^2 - 4(1)(16).
Step 5: Simplify the discriminant: D = k^2 - 64.
Step 6: Set the condition for real and distinct roots: k^2 - 64 > 0.
Step 7: Solve the inequality k^2 - 64 > 0 by factoring: (k - 8)(k + 8) > 0.
Step 8: Determine the intervals where the product (k - 8)(k + 8) is positive. This occurs when k > 8 or k < -8.

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