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

Imagine a puzzle where two mysterious numbers hold the key to a mathematical enigma. In this blog post, we'll embark on a journey to unravel the secrets behind these numbers and their intriguing relationship. Get ready to dive into the world of equations and discover the values that satisfy the given conditions!

There are two numbers whose product added to the sum of their squares is 109, and the difference of whose squares is 24. What are the two numbers?

**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.

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Through the power of algebra and careful manipulation of equations, we've cracked the code and found that the two mysterious numbers are 7 and 5. This puzzle showcases how mathematical tools can unlock hidden solutions, providing a satisfying conclusion to our mathematical adventure! Keep exploring and discovering the wonders of math!