For which value of k does the equation x² - kx + 9 = 0 have roots that are both positive? (2023)

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Q: For which value of k does the equation x² - kx + 9 = 0 have roots that are both positive? (2023)

Solution: For both roots to be positive, k must be greater than 0 and k² > 4*9 = 36, thus k > 6.

Steps: 11

Step 1: Start with the quadratic equation x² - kx + 9 = 0.
Step 2: Identify that for a quadratic equation ax² + bx + c = 0, the roots can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 3: In our equation, a = 1, b = -k, and c = 9.
Step 4: The roots will be positive if the following two conditions are met: (1) the discriminant (b² - 4ac) must be non-negative, and (2) the sum of the roots must be positive.
Step 5: Calculate the discriminant: b² - 4ac = (-k)² - 4(1)(9) = k² - 36.
Step 6: For the discriminant to be non-negative, we need k² - 36 ≥ 0, which simplifies to k² ≥ 36.
Step 7: This means k must be greater than or equal to 6 or less than or equal to -6 (k ≥ 6 or k ≤ -6).
Step 8: Next, we need to ensure that both roots are positive. The sum of the roots is given by -b/a = k/1 = k.
Step 9: For the sum of the roots (k) to be positive, k must be greater than 0.
Step 10: Combine the conditions: k must be greater than 0 and k must also be greater than 6.
Step 11: Therefore, the final condition is k > 6.