Step 5: For the roots to be real numbers, the expression inside the square root (the discriminant) must be greater than or equal to 0: 4 - 4k ≥ 0.
Step 6: Solve the inequality: 4 ≥ 4k, which simplifies to k ≤ 1.
Step 7: Now, for both roots to be negative, the sum of the roots (which is -b/a = -2/1 = -2) must be negative, and the product of the roots (which is c/a = k/1 = k) must be positive.
Step 8: Since the product of the roots (k) must be positive, we conclude that k > 0.
Step 9: Combine the conditions: k must be greater than 0 and less than or equal to 1, so the final condition is k > 0.
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 sum and product of roots and their implications on the signs of the roots.
Discriminant Analysis – Using the discriminant to ensure that the roots are real and analyzing the conditions for both roots to be negative.
