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

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Q: What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has no real roots?

Solution: For no real roots, the discriminant must be less than zero: k^2 - 4*1*9 < 0, thus k > 6.

Steps: 12

Step 1: Identify the quadratic equation, which is x^2 + kx + 9 = 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 = 9.
Step 5: Substitute the values into the discriminant formula: D = k^2 - 4*1*9.
Step 6: Simplify the expression: D = k^2 - 36.
Step 7: Set the discriminant less than zero for no real roots: k^2 - 36 < 0.
Step 8: Rearrange the inequality: k^2 < 36.
Step 9: Take the square root of both sides: |k| < 6.
Step 10: This means k is between -6 and 6: -6 < k < 6.
Step 11: However, we want the values of k that ensure no real roots, which means k must be greater than 6 or less than -6.
Step 12: Therefore, the final answer is k > 6 or k < -6.