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

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

Options:

Correct Answer: -6

Exam Year: 2023

Solution:

For both roots to be negative, k must be positive and k^2 > 4(1)(9). Thus, k > 6.

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

What is the value of k if the quadratic equation x^2 + kx + 9 = 0 has roots that are both negative? (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 quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
Step 3: Identify the coefficients: a = 1, b = k, and c = 9.
Step 4: For the roots to be real, the discriminant (b² - 4ac) must be greater than or equal to 0. Here, it is k² - 4(1)(9) = k² - 36.
Step 5: Set the discriminant greater than 0 for two distinct roots: k² - 36 > 0.
Step 6: Solve the inequality: k² > 36, which means k > 6 or k < -6.
Step 7: Since we want both roots to be negative, we need k to be positive. Therefore, we only consider k > 6.
Step 8: Conclude that the value of k must be greater than 6 for both roots to be negative.

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