Question 114234
{{{9m^2-kn^2}}} is the difference of two squares, so you are looking for something that fits the pattern:


{{{a^2-b^2=(a+b)(a-b)}}}, or put another way, {{{(a-b)=(sqrt(a)+sqrt(b))(sqrt(a)-sqrt(b))}}}


That means that the factorization of {{{9m^2-kn^2}}} is {{{(sqrt(9m^2)+sqrt(kn^2))(sqrt(9m^2)-sqrt(kn^2))}}}


But, {{{(sqrt(9m^2)+sqrt(kn^2))(sqrt(9m^2)-sqrt(kn^2))}}} must equal {{{(3m+7n)(3m-7n)}}} according to the conditions of the problem.  Therefore we can set 


{{{sqrt(kn^2)=7n}}}
{{{n*sqrt(k)=7n}}}
{{{sqrt(k)=7}}}
{{{k=49}}}


+++++++++++++++++++++++++++++++++++++++++++++++++++++++

A simpler method just to get to the answer straightaway would be to simply apply FOIL to {{{(3m+7n)(3m-7n)}}}


{{{(3m+7n)(3m-7n)=9m^2-21mn+21mn-49m^2=9m^2-49m^2}}}, and you can see by inspection that k = 49.


Hope that helps,
John