Find the value of k for which the equation x^2 + kx + 9 = 0 has roots that are b

Find the value of k for which the equation x^2 + kx + 9 = 0 has roots that are both negative.

Find the value of k for which the equation x^2 + kx + 9 = 0 has roots that are both negative.

Correct Answer: k > 6

Step 1: Understand that we need to find the value of k in the equation 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 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 negative if two conditions are met: the sum of the roots is negative and the product of the roots is positive.
Step 5: The sum of the roots (using -b/a) is -k. For this to be negative, k must be positive (k > 0).
Step 6: The product of the roots (using c/a) is 9. This is always positive since 9 is positive.
Step 7: Now, we need to ensure that the roots are both negative. For this, we need to check the discriminant (b² - 4ac).
Step 8: Calculate the discriminant: k² - 4(1)(9) = k² - 36.
Step 9: For the roots to be real and distinct, the discriminant must be greater than 0: k² - 36 > 0.
Step 10: Solve the inequality: k² > 36, which gives us k > 6 or k < -6. Since k must be positive, we take k > 6.

Quadratic Equations – Understanding the conditions for the roots of a quadratic equation based on its coefficients.
Vieta's Formulas – Using the relationships between the coefficients of a polynomial and the sums/products of its roots.
Inequalities – Applying inequalities to determine the conditions under which the roots are negative.