Comment upon the nature of roots of the quadratic equation $x^2 + 2x + k = \int_0^k |1+k| dr$ depending on the value of $k \in \mathbb{R}$.
Step-by-Step Solution
Key Concept: The absolute value function requires case-by-case analysis based on where the expression $k+t$ changes sign within the integration limits.
For the expression $D = 4 + 4\left(k + \int_0^1 |k+t| dt\right) = 4 + 4k + 4I$, evaluate $I = \int_0^1 |k+t| dt$ by cases. When $k \geq 0$: $I = k + \frac{1}{2}$, giving $D = 4 + 4k + 4(k + \frac{1}{2}) = 8 + 8k \geq 2$. When $-1 0$. When $k \leq -1$: $I = -k - \frac{1}{2}$, giving $D = 4 + 4k - 4(k + \frac{1}{2}) = 2 > 0$. Therefore $D > 0$ for all real $k$.
Correct Answer: 1,2,4