For a real number $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2 + \{\alpha\} = 21$$ can be expressed in the form $$\frac{\sqrt{a} - \sqrt{b}}{c} - d$$ where $a$, $b$, $c$, and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d$.

Let's think step by step and output the final answer within \boxed{}.
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