MathDB

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.

2