The quadratic equation x^2 - 6x + k = 0 has roots that are both positive. What is the condition on k?

1 question

1 item

Q: The quadratic equation x^2 - 6x + k = 0 has roots that are both positive. What is the condition on k?

Solution: For both roots to be positive, k must be greater than the square of half the coefficient of x: k > (6/2)^2 = 9.

Steps: 12

Step 1: Identify the quadratic equation given, which is x^2 - 6x + k = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula: roots = (-b ± √(b² - 4ac)) / (2a).
Step 3: In our equation, a = 1, b = -6, and c = k.
Step 4: For the roots to be positive, the following conditions must be satisfied: the sum of the roots must be positive and the product of the roots must be positive.
Step 5: The sum of the roots is given by -b/a, which is 6/1 = 6 (this is already positive).
Step 6: The product of the roots is given by c/a, which is k/1 = k. For the product to be positive, k must be greater than 0.
Step 7: Additionally, we need to ensure that the roots are real numbers. This requires the discriminant (b² - 4ac) to be non-negative.
Step 8: Calculate the discriminant: (-6)² - 4(1)(k) = 36 - 4k.
Step 9: Set the discriminant greater than or equal to 0 for real roots: 36 - 4k ≥ 0.
Step 10: Solve for k: 36 ≥ 4k, which simplifies to k ≤ 9.
Step 11: Combine the conditions: k must be greater than 0 (for positive product) and less than or equal to 9 (for real roots).
Step 12: The final condition for k is that it must be greater than 9 for both roots to be positive.