Where did the square roots go?

Algebra Level 4

$\dfrac a{\sqrt{a+bc}} + \dfrac b{\sqrt{b+ac}} + \dfrac c{\sqrt{c+ab}} \geq K \left( \dfrac a{a+1} + \dfrac b{b+1} + \dfrac c{c+1} \right)$

Let $$a,b$$ and $$c$$ be positive real numbers whose sum is $$1$$ and that they satisfy the inequality above for some constant $$K$$.

Let $$x$$ denote the value of $$a$$ when the equality occurs, find $$K+x$$.

Submit your answer to $$2$$ decimal places.

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