Solving for ( ab cdot cd 34 ) and ( ac - bd 19 ): A Unique Approach Using the Sum of Squares
Recently, a unique mathematical problem caught my attention. The problem presented equations that required a sum of squares approach, ensuring their resolution with precision. This article aims to dissect the solution methodically and provide clarity for future reference.
Original Equations
The equations provided were as follows: ( ab cdot cd 34 ) (E01) ( ac - bd 19 ) (E02)Step-by-Step Approach
Firstly, we need to recognize that these equations are interdependent and must be solved simultaneously. Our aim is to find the values of ( abcd ) that satisfy both these conditions. Let's start by squaring and adding both E01 and E02:Squaring and Adding Both Equations
Consider E01: ( ab cdot cd 34 )
Consider E02: ( ac - bd 19 )
Squaring and adding both E01 and E02:
[a^2d^2 b^2c^2 34^2 19^2 1517 37 times 41]
Case Analysis
Based on the factors of 1517, we need to consider the following cases:
Case - 1: When ( a^2d^2 37 ) and ( b^2c^2 41 )
[a^2d^2 37 implies ad pm 6, pm 1]
[b^2c^2 41 implies bc pm 5, pm 4]
Validating with the given equations:
Case - 1A: When ( abcd 1456 )[ E01: ab cdot cd 4 cdot 30 34 text{ (Correct)}]
[ E02: ac - bd 5 - 24 eq 19 text{ (Incorrect)}]
Case - 1B: When ( abcd 6451 )[ E01: ab cdot cd 24 cdot 5 eq 34 text{ (Incorrect)}]
Case - 1C: When ( abcd 1546 )[ E01: ab cdot cd 5 cdot 24 eq 34 text{ (Incorrect)}]
Case - 1D: When ( abcd 6541 )[ E01: ab cdot cd 30 cdot 4 34 text{ (Correct)}]
[ E02: ac - bd 24 - 5 19 text{ (Correct)}]
Therefore:
[ boxed{abcd 6541} ]
Case - 2: When ( a^2d^2 41 ) and ( b^2c^2 37 )
[a^2d^2 41 implies ad pm5, pm4]
[b^2c^2 37 implies bc pm6, pm1]
Validating with the given equations:
Case - 2A: When ( abcd 4165 )[ E01: ab cdot cd 4 cdot 30 34 text{ (Correct)}]
[ E02: ac - bd 24 - 5 19 text{ (Correct)}]
[ boxed{abcd 4165} ]
Case - 2B: When ( abcd 4615 )[ E01: ab cdot cd 24 cdot 5 eq 34 text{ (Incorrect)}]
Case - 2C: When ( abcd 5164 )[ E01: ab cdot cd 5 cdot 24 eq 34 text{ (Incorrect)}]
Case - 2D: When ( abcd 5614 )[ E01: ab cdot cd 30 cdot 4 34 text{ (Correct)}]
[ E02: ac - bd 5 - 24 eq 19 text{ (Incorrect)}]
Conclusion
From the analysis, the values of ( abcd ) that satisfy both conditions are ( 6541 ) and ( 4165 ). This method clearly demonstrates the approach to solving such algebraic problems using sums of squares, making it a valuable technique for future reference.