(A*x – B)^3 + (B*x – A)^3 = (C*x – C)^3

 
 
Note that

( \,a^2 \, x - b^2 \,)^3 \; + \; ( \,b^2 \, x - a^2 \,)^3 \; = \; ( \,c^2 \, x - c^2 \,)^3

 

If    a^2 \; + \; b^2 \; = \; c^2

and    a \; \neq \; 0 ,    b \; \neq \; 0
 
then,

Solutions:

x_1 \; = \; b^2/a^2
x_2 \; = \; a^2/b^2
x_3 \; = \; 1

x_1 \, x_2 \, x_3 \; = \; 1

 
 

How would you solve…

if    a^2 \; + \; b^2 \; \neq \; c^2

and the general equation:

( \,A \, x-B \,)^3 \; + \; ( \,B \, x-A \,)^3 \; = \; ( \,C \, x-C \,)^3
 

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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