What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots

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Question: What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots?

Options:

Correct Answer: k > 6

Solution:

The discriminant must be negative: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6, hence k > 6.

What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots?

What is the value of k for which the equation x^2 + kx + 9 = 0 has no real roots?

Correct Answer: k < -6 or k > 6

Step 1: Identify the equation given, which is x^2 + kx + 9 = 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 (D) for the equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = k, and c = 9. So, we calculate the discriminant: D = k^2 - 4*1*9.
Step 5: Simplify the discriminant: D = k^2 - 36.
Step 6: Set the discriminant less than zero for no real roots: k^2 - 36 < 0.
Step 7: Rearrange the inequality: k^2 < 36.
Step 8: Take the square root of both sides: |k| < 6.
Step 9: This means k can be between -6 and 6, so k must be greater than -6 and less than 6.

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