What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has roots that

What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has roots that are both positive? (2023)

What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has roots that are both positive? (2023)

Step 1: Understand that we have a quadratic equation in the form of x^2 + kx + 9 = 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 = k, and c = 9.
Step 4: The roots will be positive if the following conditions are met: 1) The sum of the roots (which is -b/a = -k) must be positive, and 2) The product of the roots (which is c/a = 9) must also be positive.
Step 5: For the sum of the roots to be positive, -k > 0, which means k < 0 (k must be negative).
Step 6: For the product of the roots to be positive, we need the discriminant (b² - 4ac) to be greater than 0, which ensures that the roots are real and distinct.
Step 7: Calculate the discriminant: k² - 4(1)(9) = k² - 36.
Step 8: Set the discriminant greater than 0: k² - 36 > 0.
Step 9: Solve the inequality: k² > 36, which means k < -6 or k > 6.
Step 10: Since we already established that k must be negative, we conclude that k < -6.

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