For which value of k does the equation x^2 - kx + 9 = 0 have roots that are both positive?

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Q: For which value of k does the equation x^2 - kx + 9 = 0 have roots that are both positive?

Solution: For both roots to be positive, k must be greater than 6, as the sum of the roots must be positive.

Steps: 12

Step 1: Understand that the equation x^2 - kx + 9 = 0 is a quadratic equation.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is given by -b/a.
Step 3: In our equation, a = 1, b = -k, and c = 9.
Step 4: Calculate the sum of the roots: Sum = -(-k)/1 = k.
Step 5: For both roots to be positive, the sum of the roots (k) must be greater than 0.
Step 6: Now, recall that the product of the roots is given by c/a, which is 9 in our case.
Step 7: For both roots to be positive, the product of the roots (9) must also be positive, which it is.
Step 8: However, we need to ensure that the roots are not only positive but also distinct.
Step 9: Use the discriminant (D = b^2 - 4ac) to check for distinct roots: D = (-k)^2 - 4(1)(9) = k^2 - 36.
Step 10: For the roots to be distinct, the discriminant must be greater than 0: k^2 - 36 > 0.
Step 11: Solve the inequality: k^2 > 36, which gives k > 6 or k < -6.
Step 12: Since we want both roots to be positive, we only consider k > 6.