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Algebra Level 5

Let \(P(x) = x^2 - 3x - 9\) . A real number \(x\) is chosen at random from the interval \(5 \le x \le 15\) . The probability that \(\left \lfloor\sqrt{P(x)}\right \rfloor = \sqrt{P(\lfloor x \rfloor)}\) is equal to \(\dfrac{\sqrt{a} + \sqrt{b} + \sqrt{c} - d}{e}\) , where \(a, b, c, d,\) and \(e\) are positive integers. Find \(a + b + c + d + e\).

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