Solving $(m,n)$ for $m^2-2mn+14n^2=217$

  • MHB
  • Thread starter Albert1
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In summary, the equation that needs to be solved for $(m,n)$ is $(m^2-2mn+14n^2=217)$. The variables in the equation are $m$ and $n$, and the method for solving this type of equation is factoring or using the quadratic formula. This equation can have multiple solutions for $(m,n)$, and solving it can help in understanding patterns and relationships between variables, which is important in scientific research in fields such as physics, chemistry, and engineering.
  • #1
Albert1
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$m^2-2mn+14n^2=217$
$m,n\in N$
find solution(s) of $(m,n)$
 
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  • #2
Albert said:
$m^2-2mn+14n^2=217$
$m,n\in N$
find solution(s) of $(m,n)$
we have

$(m-n)^2 + 13n^2 = 217$
so $13n^2 < 127$ or $n^2 < 17 ( 13 * 17 = 221)$
trying n = 1,2,3,4,5 we get
n = 3 and m-n = 10 so m = 13 , n =3 is the solution and no other
 
Last edited:
  • #3
kaliprasad said:
we have

$(m-n)^2 + 13n^3 = 217$
so $13n^2 < 127$ or $n^2 < 17 ( 13 * 17 = 221)$
trying n = 1,2,3,4,5 we get
n = 3 and m-n = 10 so m = 13 , n =3 is the solution and no other
sorry! have some others
 
  • #4
Albert said:
sorry! have some others

Yes I missed one more n=4, n-m = 3 giving m = 7, n= 4
 
  • #5
kaliprasad said:
Yes I missed one more n=4, n-m = 3 giving m = 7, n= 4
still one missing
 
  • #6
Albert said:
still one missing

I should have taken |m-n| =3 giving m = 1 and n = 4
 

FAQ: Solving $(m,n)$ for $m^2-2mn+14n^2=217$

What is the equation that needs to be solved for $(m,n)$?

The equation that needs to be solved is $(m^2-2mn+14n^2=217)$.

What are the variables in the equation?

The variables in the equation are $m$ and $n$.

What is the method for solving this type of equation?

The method for solving this type of equation is factoring or using the quadratic formula.

Can this equation have multiple solutions?

Yes, this equation can have multiple solutions for $(m,n)$.

What is the importance of solving this equation in terms of scientific research?

Solving this equation can help in understanding patterns and relationships between variables, which can be useful in various scientific fields such as physics, chemistry, and engineering.

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