Solving the Number Puzzle: A Quadratic Equations and Algebraic Manipulation Puzzle

The question at hand involves two numbers, whose product when added to the sum of their squares equals 109, and the difference of their squares is 24. To solve this puzzle, we will use algebraic manipulation and quadratic equations.

Understanding the Problem

The given conditions can be translated into the following equations:

Product of the numbers plus the sum of their squares equals 109:

a2b2 ab 109

Difference of the squares of the numbers equals 24:

a2 - b2 24

Step-by-Step Solution

To solve for the numbers, we will first introduce variables. Let the smaller number be x and the larger be y.

Equation 1: x2y2 xy 109

Equation 2: y2 - x2 24

Let's manipulate the equations to find the solution step by step.

Combining the Equations

Add both Equations 1 and 2:

(x2y2 xy) (y2 - x2) 109 24

this simplifies to: 2y2 xy 133

Divide the entire equation by 2:

y2 (1/2)xy 66.5

Use the difference of squares to find the value of y:

y2 - x2 24 can be rewritten as 24 (y x)(y - x)

Let's solve for y and x by trial and error, which is often easier than attempting to solve quadratic equations directly.

Using Trial and Error

I started by considering square numbers with a difference of 24. The square numbers 49 and 25 have a difference of 24, and their square roots are 7 and 5, respectively. These numbers satisfy both conditions:

y2 - x2 24 is satisfied by (72 - 52) 49 - 25 24

x2y2 xy 109 is satisfied by (5272 57) (2525 35) 109

Thus, the two numbers 5 and 7 satisfy both conditions.

Conclusion

The solution to the puzzle is the pair (5, 7). If you check the values, they fit both equations:

5272 57 109

72 - 52 49 - 25 24

This method of trial and error is often more straightforward when dealing with complex algebraic equations.

Related Keywords and Search Terms

Quadratic Equations, Algebraic Manipulation, Puzzle Solution