<p>The value of \(k\) for which both roots of the equation \(4x^2 - 2x + k = 0\) are completely in \((-1, 1)\), may be equal to:</p>
Step-by-Step Solution
Key Concept: Use conditions on discriminant, vertex location, and function values at interval endpoints.
<p>For both roots to lie in $(-1, 1)$, we need: (1) Discriminant $\geq 0$: $4 - 16k \geq 0 \Rightarrow k \leq 1/4$; (2) Vertex inside $(-1, 1)$: vertex at $x = 1/4 \in (-1, 1)$ ✓; (3) $f(-1) > 0$: $4 + 2 + k > 0 \Rightarrow k > -6$; (4) $f(1) > 0$: $4 - 2 + k > 0 \Rightarrow k > -2$. Combined: $-2 < k \leq 1/4$. Check options: $k = -1 \in (-2, 1/4]$ ✓, $k = 0 \in (-2, 1/4]$ ✓.</p>
Correct Answer: A, B