Question

The sum of a number ,its square and its square root is 22 . What is the number ?

Answer 1

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 \( \boxed{4} \).