Let the number be \( x \). According to the problem, we can set up the equation:
\[ x + x^2 + \sqrt{x} = 22 \]
To solve this equation, we will rearrange it:
\[ x^2 + x + \sqrt{x} - 22 = 0 \]
This is a complicated equation due to the square root term. Let's isolate \( \sqrt{x} \):
\[ \sqrt{x} = 22 - x - x^2 \]
Now we can square both sides to eliminate the square root:
\[ x = (22 - x - x^2)^2 \]
Expanding the right-hand side:
\[ x = (22 - x - x^2)(22 - x - x^2) \] \[ = (22 - x - x^2)^2 \] Expanding further:
\[ (22 - x - x^2)(22 - x - x^2) = 484 - 44x - 44x^2 + x^2 + 2x^3 + x^4 \]
This is pretty complex, so let's check for possible integer solutions. We can try substituting some reasonable values for \( x \) to see if they satisfy the original equation.
Let's test \( x = 9 \):
\[ 9 + 9^2 + \sqrt{9} = 9 + 81 + 3 = 93 \quad \text{(too high)} \]
Now let's try \( x = 4 \):
\[ 4 + 4^2 + \sqrt{4} = 4 + 16 + 2 = 22 \quad \text{(this works!)} \]
Therefore, the number is:
\[ \boxed{4} \]
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