If the equation x^2 + 5x + k = 0 has roots that are both negative, what is the c

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Question: If the equation x^2 + 5x + k = 0 has roots that are both negative, what is the condition for k?

Options:

Correct Answer: k > 0

Solution:

For both roots to be negative, the sum of the roots (which is -5) must be negative, and the product (k) must be positive. Thus, k > 0.

If the equation x^2 + 5x + k = 0 has roots that are both negative, what is the condition for k?

If the equation x^2 + 5x + k = 0 has roots that are both negative, what is the condition for k?

Step 1: Identify the equation given, which is x^2 + 5x + k = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots is given by -b/a and the product of the roots is given by c/a.
Step 3: In our equation, a = 1, b = 5, and c = k.
Step 4: Calculate the sum of the roots: Sum = -b/a = -5/1 = -5.
Step 5: Since the sum of the roots is -5, which is negative, this condition is satisfied for both roots to be negative.
Step 6: Now, calculate the product of the roots: Product = c/a = k/1 = k.
Step 7: For both roots to be negative, the product of the roots (k) must be positive.
Step 8: Therefore, the condition for k is that k must be greater than 0.

Quadratic Equations – Understanding the properties of roots of quadratic equations, specifically the relationships between coefficients and roots.
Vieta's Formulas – Applying Vieta's formulas to determine conditions on the coefficients based on the nature of the roots.
Inequalities – Using inequalities to establish conditions for the signs of the roots based on their sum and product.