The discussion revolves around solving two simultaneous linear equations involving variables c and d, expressed in terms of constants derived from square roots. The equations are set up as 2 = c(1+sqrt(2)) + d(1-sqrt(2)) and 5 = c(1+sqrt(2))^2 + d(1-sqrt(2))^2.
Some participants have provided guidance on how to approach the problem, suggesting specific algebraic manipulations. There is a recognition of previous mistakes in the original poster's attempts, and one participant confirms achieving the correct answer through a suggested method.
There is an emphasis on understanding the equations as linear and the need for clear working steps to identify errors in reasoning. The original poster's previous attempts are noted as having led to incorrect conclusions, prompting a reevaluation of their approach.
HallsofIvy said:You understand that these are just linear equations, don't you: Ac+ Bd= 2 and A^2c+ B^2d= 5 with A= 1+ \sqrt{2} and B= 1- \sqrt{2}. Yes, you could solve the first equation for c, c= (2- Bd)/A, and put that into the second equation: A^2[(2- Bd)/A)+ B^2d= A(2- Bd)+ B^2d= 2A- ABd+B^2d= 5 so (B^2- AB)d= 5- 2A and d= (5- 2A)/(B^2- AB)
Personally, I would have multiplied the first equation by A to get A^2c+ ABd= 2A and then subtract that from the second equation: (B^2- AB)d= 5- 2A which immediately gives d= (5- 2A)/(B^2- AB). Similarly, multiply the first equation by B to get ABc+ B^2d= 2B and subtract that from the second equation to get (A^2- AB)c= 5- 2B so that c= (5- 2B)/(A^2- AB).
Now put A= 1+\sqrt{2} and B= 1- \sqrt{2}.