Let the number be \( x \). According to the problem, we have the equation:
\[ x + x^2 + \sqrt{x} = 22 \]
To solve this equation, we can first rearrange it as:
\[ x^2 + x + \sqrt{x} - 22 = 0 \]
This is a non-linear equation due to the presence of the square root. To make the equation easier to handle, we can substitute \( y = \sqrt{x} \), which means \( x = y^2 \). Substituting this into the equation gives us:
\[ (y^2)^2 + y^2 + y - 22 = 0 \] \[ y^4 + y^2 + y - 22 = 0 \]
Now, we will look for possible rational roots or attempt to find roots using numerical or graphical methods. However, this degree of polynomial could also be complicated; it might help to try specific values for \( y \).
Trying \( y = 2 \):
\[ 2^4 + 2^2 + 2 - 22 = 16 + 4 + 2 - 22 = 0 \]
Thus, \( y = 2 \) is a root. Since \( y = \sqrt{x} \):
\[ \sqrt{x} = 2 \implies x = 2^2 = 4 \]
To confirm, we will substitute \( x = 4 \) back into the original equation:
\[ 4 + 4^2 + \sqrt{4} = 4 + 16 + 2 = 22 \]
The left side equals 22, thus confirming our solution is correct.
Therefore, the number is \( \boxed{4} \).
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