An Algebra problem by Aira Thalca
How many pairs of non-negative integers with satisfy the equation ?
The answer is 7.
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1 solution
Starting from the given equation, we obtain the equivalent equation
5 x 2 − 4 x y + 2 x + y 2 = 6 2 4
5 x 2 − 4 x y + 2 x + y 2 + 1 = 6 2 5
4 x 2 − 4 x y + y 2 + x 2 + 2 x + 1 = 6 2 5
( 2 x − y ) 2 + ( x + 1 ) 2 = 2 5 2
Since x a n d y are both integers, then the left side of the given equations is the perfect square
From Pythagorean Theorem, the list of x , y are
( 2 4 , 4 8 ) , ( 2 3 , 3 9 ) , ( 2 3 , 5 3 ) , ( 6 , 3 6 ) , ( 1 9 , 2 3 ) , ( 1 9 , 5 3 ) , ( 1 4 , 4 8 )