Answer:
m = sqrt(4/(3 n) + 1/(9 n^2)) + 1/(3 n) or m = 1/(3 n) - sqrt(4/(3 n) + 1/(9 n^2))
Explanation:
Solve for m:
4 n + 2 m n = 3 m^2 n^2
Subtract 3 m^2 n^2 from both sides:
4 n + 2 m n - 3 m^2 n^2 = 0
Divide both sides by -3 n^2:
m^2 - 4/(3 n) - (2 m)/(3 n) = 0
Add 4/(3 n) to both sides:
m^2 - (2 m)/(3 n) = 4/(3 n)
Add 1/(9 n^2) to both sides:
m^2 + 1/(9 n^2) - (2 m)/(3 n) = 4/(3 n) + 1/(9 n^2)
Write the left hand side as a square:
(m - 1/(3 n))^2 = 4/(3 n) + 1/(9 n^2)
Take the square root of both sides:
m - 1/(3 n) = sqrt(4/(3 n) + 1/(9 n^2)) or m - 1/(3 n) = -sqrt(4/(3 n) + 1/(9 n^2))
Add 1/(3 n) to both sides:
m = sqrt(4/(3 n) + 1/(9 n^2)) + 1/(3 n) or m - 1/(3 n) = -sqrt(4/(3 n) + 1/(9 n^2))
Add 1/(3 n) to both sides:
Answer: m = sqrt(4/(3 n) + 1/(9 n^2)) + 1/(3 n) or m = 1/(3 n) - sqrt(4/(3 n) + 1/(9 n^2))