Probability of Root Placement

Algebra Level 4

\(k\) is uniformly chosen from the interval \([ -5, 5] \). Let \(p\) be the probability that the quadratic \( f(x) = x^2 + kx + 1 \) has both roots between -2 and 4. What is the value of \( \lfloor 1000 p \rfloor \)?

Details and assumptions

Greatest Integer Function: \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer less than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -3.4 \rfloor = -4\).

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