What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has no real r

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Question: What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has no real roots?

Options:

Correct Answer: -4

Solution:

The discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => k < 8 and k > -8.

What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has no real roots?

What is the value of k if the quadratic equation x^2 + kx + 16 = 0 has no real roots?

Step 1: Identify the quadratic equation, which is x^2 + kx + 16 = 0.
Step 2: Recall that a quadratic equation has no real roots if its discriminant is less than zero.
Step 3: The discriminant (D) for the equation ax^2 + bx + c is given by the formula 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 value of k must be between -8 and 8 (not including -8 and 8).

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