What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots tha

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Question: What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct? (2019)

Options:

Correct Answer: -5

Exam Year: 2019

Solution:

For real and distinct roots, the discriminant must be positive: k^2 - 4(1)(16) > 0. Thus, k^2 > 64, leading to k < -8 or k > 8.

What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct? (2019)

What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has roots that are both real and distinct? (2019)

Step 1: Identify the quadratic equation given, which is x^2 + kx + 16 = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant is given by the formula D = b^2 - 4ac.
Step 3: In our equation, a = 1, b = k, and c = 16.
Step 4: Substitute the values into the discriminant formula: D = k^2 - 4(1)(16).
Step 5: Simplify the expression: D = k^2 - 64.
Step 6: For the roots to be real and distinct, the discriminant must be greater than 0: k^2 - 64 > 0.
Step 7: Rearrange the inequality: k^2 > 64.
Step 8: Take the square root of both sides: k < -8 or k > 8.
Step 9: Conclude that the values of k must be either less than -8 or greater than 8 for the roots to be real and distinct.

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