The quadratic equation x^2 + 4x + k = 0 has roots that are both negative. What is the condition on k?

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Q: The quadratic equation x^2 + 4x + k = 0 has roots that are both negative. What is the condition on k?

Solution: For both roots to be negative, the sum of roots (4) must be positive and the product (k) must be positive, hence k > 0.

Steps: 8

Step 1: Identify the quadratic equation, which is x^2 + 4x + 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 = 4, and c = k.
Step 4: Calculate the sum of the roots: Sum = -b/a = -4/1 = -4.
Step 5: For both roots to be negative, the sum of the roots must be positive. Since -4 is not positive, we need to check the product of the roots.
Step 6: 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 must also be positive. Therefore, k must be greater than 0.
Step 8: Conclude that the condition on k for both roots to be negative is k > 0.