(3m+5n)(3m+5n)-(2n)(2n)=0

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Solution for (3m+5n)(3m+5n)-(2n)(2n)=0 equation: 


Simplifying
(3m + 5n)(3m + 5n) + -1(2n)(2n) = 0

Multiply (3m + 5n) * (3m + 5n)
(3m * (3m + 5n) + 5n * (3m + 5n)) + -1(2n)(2n) = 0
((3m * 3m + 5n * 3m) + 5n * (3m + 5n)) + -1(2n)(2n) = 0

Reorder the terms:
((15mn + 9m2) + 5n * (3m + 5n)) + -1(2n)(2n) = 0
((15mn + 9m2) + 5n * (3m + 5n)) + -1(2n)(2n) = 0
(15mn + 9m2 + (3m * 5n + 5n * 5n)) + -1(2n)(2n) = 0
(15mn + 9m2 + (15mn + 25n2)) + -1(2n)(2n) = 0

Reorder the terms:
(15mn + 15mn + 9m2 + 25n2) + -1(2n)(2n) = 0

Combine like terms: 15mn + 15mn = 30mn
(30mn + 9m2 + 25n2) + -1(2n)(2n) = 0

Remove parenthesis around (2n)
30mn + 9m2 + 25n2 + -1 * 2n(2n) = 0

Remove parenthesis around (2n)
30mn + 9m2 + 25n2 + -1 * 2n * 2n = 0

Reorder the terms for easier multiplication:
30mn + 9m2 + 25n2 + -1 * 2 * 2n * n = 0

Multiply -1 * 2
30mn + 9m2 + 25n2 + -2 * 2n * n = 0

Multiply -2 * 2
30mn + 9m2 + 25n2 + -4n * n = 0

Multiply n * n
30mn + 9m2 + 25n2 + -4n2 = 0

Combine like terms: 25n2 + -4n2 = 21n2
30mn + 9m2 + 21n2 = 0

Solving
30mn + 9m2 + 21n2 = 0

Solving for variable 'm'.

Factor out the Greatest Common Factor (GCF), '3'.
3(10mn + 3m2 + 7n2) = 0

Factor a trinomial.
3((m + n)(3m + 7n)) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(m + n)' equal to zero and attempt to solve:

Simplifying
m + n = 0

Solving
m + n = 0

Move all terms containing m to the left, all other terms to the right.

Add '-1n' to each side of the equation.
m + n + -1n = 0 + -1n

Combine like terms: n + -1n = 0
m + 0 = 0 + -1n
m = 0 + -1n
Remove the zero:
m = -1n

Simplifying
m = -1n

Subproblem 2
Set the factor '(3m + 7n)' equal to zero and attempt to solve:

Simplifying
3m + 7n = 0

Solving
3m + 7n = 0

Move all terms containing m to the left, all other terms to the right.

Add '-7n' to each side of the equation.
3m + 7n + -7n = 0 + -7n

Combine like terms: 7n + -7n = 0
3m + 0 = 0 + -7n
3m = 0 + -7n
Remove the zero:
3m = -7n

Divide each side by '3'.
m = -2.333333333n

Simplifying
m = -2.333333333n
Solution
m = {-1n, -2.333333333n}

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