What is the value of k if the equation x^2 - kx + 9 = 0 has roots that are both positive?

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Q: What is the value of k if the equation x^2 - kx + 9 = 0 has roots that are both positive?

Solution: For both roots to be positive, k must be greater than 6 (sum of roots) and k^2 - 36 > 0 (discriminant). Thus, k > 6 and k < 12.

Steps: 14

Step 1: Identify the equation given, which is x^2 - kx + 9 = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is given by -b/a and the product of the roots is c/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. Therefore, k > 0.
Step 6: Calculate the product of the roots: Product = 9/1 = 9.
Step 7: For both roots to be positive, the product of the roots must also be positive. Since 9 is positive, this condition is satisfied.
Step 8: Now, we need to ensure that the discriminant (D) of the equation is positive for the roots to be real and distinct.
Step 9: The discriminant D is given by D = b^2 - 4ac = (-k)^2 - 4(1)(9) = k^2 - 36.
Step 10: Set the discriminant greater than 0: k^2 - 36 > 0.
Step 11: Solve the inequality: k^2 > 36, which gives k > 6 or k < -6. Since we want positive roots, we only consider k > 6.
Step 12: Now, we also need to ensure that the sum of the roots (k) is less than the product of the roots (9) for both roots to be positive.
Step 13: Set the inequality: k < 12.
Step 14: Combine the two inequalities: k > 6 and k < 12.