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

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

Options:

Correct Answer: -6

Solution:

For both roots to be negative, k must be negative and greater than the square root of the discriminant. Thus, k < -6.

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

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

Step 1: Understand that we have a quadratic equation in the form of x^2 + kx + 9 = 0.
Step 2: Identify that the roots of the equation are the values of x that make the equation equal to zero.
Step 3: Recall that for a quadratic equation ax^2 + bx + c = 0, the roots can be found using the formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 4: In our equation, a = 1, b = k, and c = 9.
Step 5: Calculate the discriminant, which is b² - 4ac. Here, it is k² - 4(1)(9) = k² - 36.
Step 6: For the roots to be real numbers, the discriminant must be greater than or equal to zero: k² - 36 ≥ 0.
Step 7: Solve the inequality k² - 36 ≥ 0. This gives us k ≤ -6 or k ≥ 6.
Step 8: Since we want both roots to be negative, we need to ensure that k is negative.
Step 9: Therefore, we focus on the part of the solution where k ≤ -6, which means k must be less than -6.
Step 10: Conclude that for both roots to be negative, k must be less than -6.

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