What is the value of k for which the equation x^2 + kx + 16 = 0 has no real root

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Question: What is the value of k for which the equation x^2 + kx + 16 = 0 has no real roots? (2021)

Options:

Correct Answer: -8

Exam Year: 2021

Solution:

The discriminant must be less than zero. Thus, k^2 - 4*1*16 < 0 leads to k^2 < 64, giving k < 8 and k > -8.

What is the value of k for which the equation x^2 + kx + 16 = 0 has no real roots? (2021)

What is the value of k for which the equation x^2 + kx + 16 = 0 has no real roots? (2021)

Step 1: Identify the equation given, which is x^2 + kx + 16 = 0.
Step 2: Understand that for a quadratic equation to have no real roots, the discriminant must be less than zero.
Step 3: The discriminant formula for the equation ax^2 + bx + c is given by D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = k, and c = 16.
Step 5: Substitute the values into the discriminant formula: D = k^2 - 4*1*16.
Step 6: Simplify the expression: D = k^2 - 64.
Step 7: Set the discriminant less than zero for no real roots: k^2 - 64 < 0.
Step 8: Rearrange the inequality: k^2 < 64.
Step 9: Take the square root of both sides: -8 < k < 8.
Step 10: Conclude that the values of k for which the equation has no real roots are k < 8 and k > -8.

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