If the quadratic equation x^2 + 6x + k = 0 has roots that are both negative, wha

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Question: If the quadratic equation x^2 + 6x + k = 0 has roots that are both negative, what is the condition for k?

Options:

Correct Answer: k > 9

Solution:

For both roots to be negative, k must be greater than the square of half the coefficient of x, hence k > 9.

If the quadratic equation x^2 + 6x + k = 0 has roots that are both negative, what is the condition for k?

If the quadratic equation x^2 + 6x + k = 0 has roots that are both negative, what is the condition for k?

Correct Answer: k > 9

Step 1: Identify the quadratic equation, which is x^2 + 6x + k = 0.
Step 2: Recognize that the roots of the equation can be found using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a.
Step 3: In our equation, a = 1, b = 6, and c = k.
Step 4: For the roots to be negative, the sum of the roots (which is -b/a) must be negative. Here, -b/a = -6/1 = -6, which is already negative.
Step 5: Next, we need to ensure that the product of the roots (which is c/a) is positive. The product of the roots is k/1 = k.
Step 6: For the product of the roots to be positive, k must be greater than 0 (k > 0).
Step 7: Additionally, we need to ensure that both roots are less than -3 (since the sum of the roots is -6, both must be less than half of -6). This means k must also satisfy the condition derived from the vertex of the parabola.
Step 8: The vertex of the parabola occurs at x = -b/(2a) = -6/2 = -3. For both roots to be negative, k must be greater than the square of -3, which is 9.
Step 9: Therefore, the final condition for k is k > 9.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the conditions for the nature of roots based on coefficients.
Roots of Quadratic Equations – Applying Vieta's formulas to determine conditions on coefficients based on the signs of the roots.
Discriminant Analysis – Using the discriminant to analyze the nature of roots (real and distinct, real and equal, or complex).