Added to the sum of their squares is 109 -Math Riddle

Answer: 5 and 7. 

Explanation :  

We have two conditions:

The product of two numbers added to the sum of their squares is 109, which can be represented as:
xy + x² + y² = 109.

The difference of their squares is 24, which can be represented as:
x² - y² = 24.

Now, let's correctly solve these equations:

From Equation 2 (x² - y² = 24), you found that x² = 49, which implies x = 7. This part is correct.

However, the step where we substitute x = 7 into Equation 1 is where the error occurred:

(7)y + (7)² + y² = 109
7y + 49 + y² = 109

Now, let's correct the calculation:

7y + 49 + y² = 109

To isolate the terms involving y, subtract 49 from both sides:

7y + y² = 109 - 49

Now, it should be:

7y + y² = 60

Next, rearrange the terms:

y² + 7y = 60

To solve for y, let's rewrite the equation in the form of a quadratic equation:

y² + 7y - 60 = 0

Now, you can factor the quadratic equation:

(y + 12)(y - 5) = 0

Setting each factor equal to zero gives us two possible values for y:

y + 12 = 0, which leads to y = -12.
y - 5 = 0, which leads to y = 5.
So, there are two possible pairs of numbers that satisfy the given conditions:

x = 7 and y = -12
x = 7 and y = 5
Both pairs make the equations true, and the original problem has two valid solutions.