For the equation x² + 6x + k = 0 to have real roots, what must be the minimum value of k? (2023)

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Q: For the equation x² + 6x + k = 0 to have real roots, what must be the minimum value of k? (2023)

Solution: The discriminant must be non-negative: 6² - 4*1*k ≥ 0, thus k ≤ 9.

Steps: 9

Step 1: Identify the equation given, which is x² + 6x + k = 0.
Step 2: Recognize that for a quadratic equation to have real roots, the discriminant must be non-negative.
Step 3: The discriminant (D) for the equation ax² + bx + c = 0 is calculated using the formula D = b² - 4ac.
Step 4: In our equation, a = 1, b = 6, and c = k. So, we substitute these values into the discriminant formula: D = 6² - 4*1*k.
Step 5: Calculate 6², which is 36. Now we have D = 36 - 4k.
Step 6: Set the discriminant greater than or equal to zero for real roots: 36 - 4k ≥ 0.
Step 7: Rearrange the inequality to find k: 36 ≥ 4k, or 4k ≤ 36.
Step 8: Divide both sides of the inequality by 4: k ≤ 9.
Step 9: Conclude that the minimum value of k for the equation to have real roots is k = 9.