Let's denote the two numbers as \( x \) and \( y \). According to the problem, we have the following two equations:
- \( x - y = 2 \)
- \( x^2 + y^2 = 244 \)
From the first equation, we can express \( x \) in terms of \( y \):
\[ x = y + 2 \]
Now, we can substitute this expression for \( x \) into the second equation:
\[ (y + 2)^2 + y^2 = 244 \]
Expanding the first term:
\[ (y^2 + 4y + 4) + y^2 = 244 \]
Combining like terms:
\[ 2y^2 + 4y + 4 = 244 \]
Subtracting 244 from both sides gives:
\[ 2y^2 + 4y + 4 - 244 = 0 \]
Simplifying further:
\[ 2y^2 + 4y - 240 = 0 \]
Now, we can divide the entire equation by 2 to simplify:
\[ y^2 + 2y - 120 = 0 \]
Next, we will factor or use the quadratic formula to solve for \( y \). Factoring, we want two numbers that multiply to \( -120 \) and add to \( 2 \). These numbers are \( 12 \) and \( -10 \):
\[ (y + 12)(y - 10) = 0 \]
Setting each factor to zero gives:
\[ y + 12 = 0 \quad \Rightarrow \quad y = -12 \] \[ y - 10 = 0 \quad \Rightarrow \quad y = 10 \]
Now we can find the corresponding values for \( x \):
-
If \( y = -12 \): \[ x = y + 2 = -12 + 2 = -10 \]
-
If \( y = 10 \): \[ x = y + 2 = 10 + 2 = 12 \]
So the pairs of numbers are:
- \( (-10, -12) \)
- \( (12, 10) \)
In conclusion, the two numbers are:
\[ \boxed{(12, 10) \text{ or } (-10, -12)} \]