The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What i

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Question: The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)

Options:

Correct Answer: k > 9

Exam Year: 2020

Solution:

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

The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)

The quadratic equation x^2 + 6x + k = 0 has roots that are both negative. What is the condition for k? (2020)

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 formula for the roots of a quadratic equation.
Step 3: For the roots to be negative, we need to consider the properties of the roots based on the coefficients.
Step 4: The sum of the roots (which is -b/a) must be negative. Here, b = 6, so the sum of the roots is -6/a, which is negative.
Step 5: The product of the roots (which is c/a) must be positive. Here, c = k, so the product of the roots is k/a, which must be positive.
Step 6: Since a = 1 (the coefficient of x^2), we need k > 0 for the product of the roots to be positive.
Step 7: Additionally, we need to ensure that the roots are both negative. This requires that k must be greater than the square of half the coefficient of x.
Step 8: Calculate half of the coefficient of x: half of 6 is 3. The square of 3 is 9.
Step 9: Therefore, for both roots to be negative, k must be greater than 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 the conditions under which the roots of a quadratic equation are negative.
Discriminant and Vertex – Using the vertex form and discriminant to analyze the conditions for the roots of a quadratic equation.