SOLUTION: Find a value for k so that 9m squared - kn squared will have the factors 3m + 7n and 3m - 7n

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Question 114234: Find a value for k so that 9m squared - kn squared will have the factors 3m + 7n and 3m - 7n
Found 2 solutions by stanbon, solver91311:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Find a value for k so that 9m squared - kn squared will have the factors 3m + 7n and 3m - 7n
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If they are factors:
9m^2-kn^2 = (3m+7n)(3m-7n)
9m^2-kn^2 = 9m^2-49n^2
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-k = -49
k = 49
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Cheers,
Stan H.

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
9m%5E2-kn%5E2 is the difference of two squares, so you are looking for something that fits the pattern:

a%5E2-b%5E2=%28a%2Bb%29%28a-b%29, or put another way, %28a-b%29=%28sqrt%28a%29%2Bsqrt%28b%29%29%28sqrt%28a%29-sqrt%28b%29%29

That means that the factorization of 9m%5E2-kn%5E2 is %28sqrt%289m%5E2%29%2Bsqrt%28kn%5E2%29%29%28sqrt%289m%5E2%29-sqrt%28kn%5E2%29%29

But, %28sqrt%289m%5E2%29%2Bsqrt%28kn%5E2%29%29%28sqrt%289m%5E2%29-sqrt%28kn%5E2%29%29 must equal %283m%2B7n%29%283m-7n%29 according to the conditions of the problem. Therefore we can set

sqrt%28kn%5E2%29=7n
n%2Asqrt%28k%29=7n
sqrt%28k%29=7
k=49

+++++++++++++++++++++++++++++++++++++++++++++++++++++++
A simpler method just to get to the answer straightaway would be to simply apply FOIL to %283m%2B7n%29%283m-7n%29

%283m%2B7n%29%283m-7n%29=9m%5E2-21mn%2B21mn-49m%5E2=9m%5E2-49m%5E2, and you can see by inspection that k = 49.

Hope that helps,
John