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Let's assume that the two numbers are x and y.

We have two equations based on the given information:

1. x^2 + y^2 = 277

2. x^2 - y^2 = 115

To solve this system of equations, we can use the method of substitution.

First, let's rearrange equation 2 to solve for x^2:

x^2 = y^2 + 115

Substituting this value for x^2 in equation 1, we get:

(y^2 + 115) + y^2 = 277

2y^2 + 115 = 277

Next, we can simplify the equation:

2y^2 = 277 - 115

2y^2 = 162

Now, divide both sides by 2:

y^2 = 162 / 2

y^2 = 81

Taking the square root of both sides, we find:

y = √81

Since the square root of 81 is 9 or -9, we have two options for the value of y.

Case 1: y = 9

If y = 9, we can substitute this value back into equation 2 to solve for x:

x^2 - 9^2 = 115

x^2 - 81 = 115

x^2 = 115 + 81

x^2 = 196

x = ±√196

x = ±14

Therefore, one possible solution is x = 14, y = 9.

Case 2: y = -9

If y = -9, we can repeat the same steps to find x:

x^2 - (-9)^2 = 115

x^2 - 81 = 115

x^2 = 115 + 81

x^2 = 196

x = ±√196

x = ±14

Therefore, the other possible solution is x = -14, y = -9.

So, the two pairs of numbers that satisfy the given conditions are (14, 9) and (-14, -9).

Answer:

Let's assume the two numbers as x and y.

According to the given information, we have two equations:

1. x^2 + y^2 = 277 (Equation 1)

2. x^2 - y^2 = 115 (Equation 2)

To solve these equations, we can use a method called substitution.

From Equation 2, we can express x^2 in terms of y^2:

x^2 = y^2 + 115

Substituting this expression into Equation 1, we have:

(y^2 + 115) + y^2 = 277

2y^2 + 115 = 277

2y^2 = 277 - 115

2y^2 = 162

y^2 = 162 / 2

y^2 = 81

y = √81

y = 9

Substituting the value of y = 9 into Equation 2, we can find x:

x^2 = 9^2 + 115

x^2 = 81 + 115

x^2 = 196

x = √196

x = 14

Therefore, the two numbers are x = 14 and y = 9.