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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