SOLUTION: For the equation (a-b)^2+a^2=25, a and b are integers and a ≥ 0. List the ordered pairs that occur.

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Question 1105718: For the equation (a-b)^2+a^2=25, a and b are integers and a ≥ 0. List the ordered pairs that occur.
Found 2 solutions by Boreal, Edwin McCravy:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
(a-b)^2=(25-a^2)
(a-b)=sqrt(25-a^2)
-b=-a+sqrt(25-a^2)
b=a-sqrt(25-a^2)
a can't be greater than 5, but a=5 b=0 works
a=4 b=1 works
a=3 and b=-1 works
a=2 and b can't be an integer
nor can a=1
a=0, b=5 works
(5, 0)
(0, -5)
(4, 1)
(3, -1)

Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
%28a-b%29%5E2%2Ba%5E2=25

%28a-b%29%5E2%2Ba%5E2=5%5E2

We know the 3,4,5 right triangle 

That is, the Pythagorean triple (3,4,5)
where
3%5E2%2B4%5E2+=+5%5E2
4%5E2%2B3%5E2+=+5%5E2

and we know

0%5E2%2B5%5E2+=+5%5E2
5%5E2%2B0%5E2+=+5%5E2

So we have

system%28abs%28a-b%29=3%2Ca=4%29 

which has solutions (a,b) = (4,1) or (4,7) 

system%28abs%28a-b%29=4%2Ca=3%29

which has solutions (a,b) = (3,-1) or (3,7) 

system%28abs%28a-b%29=5%2Ca=0%29

which has solutions (a,b) = (0,5) or (5,0) 

system%28abs%28a-b%29=0%2Ca=5%29

which has only solution (a,b) = (5,5) 

Edwin