3(1+4n)2(5n-3)=25

Solution for 3(1+4n)2(5n-3)=25 equation:

Simplifying
3(1 + 4n) * 2(5n + -3) = 25

Reorder the terms:
3(1 + 4n) * 2(-3 + 5n) = 25

Reorder the terms for easier multiplication:
3 * 2(1 + 4n)(-3 + 5n) = 25

Multiply 3 * 2
6(1 + 4n)(-3 + 5n) = 25

Multiply (1 + 4n) * (-3 + 5n)
6(1(-3 + 5n) + 4n * (-3 + 5n)) = 25
6((-3 * 1 + 5n * 1) + 4n * (-3 + 5n)) = 25
6((-3 + 5n) + 4n * (-3 + 5n)) = 25
6(-3 + 5n + (-3 * 4n + 5n * 4n)) = 25
6(-3 + 5n + (-12n + 20n2)) = 25

Combine like terms: 5n + -12n = -7n
6(-3 + -7n + 20n2) = 25
(-3 * 6 + -7n * 6 + 20n2 * 6) = 25
(-18 + -42n + 120n2) = 25

Solving
-18 + -42n + 120n2 = 25

Solving for variable 'n'.

Reorder the terms:
-18 + -25 + -42n + 120n2 = 25 + -25

Combine like terms: -18 + -25 = -43
-43 + -42n + 120n2 = 25 + -25

Combine like terms: 25 + -25 = 0
-43 + -42n + 120n2 = 0

Begin completing the square.  Divide all terms by
120 the coefficient of the squared term: 

Divide each side by '120'.
-0.3583333333 + -0.35n + n2 = 0.0

Move the constant term to the right:

Add '0.3583333333' to each side of the equation.
-0.3583333333 + -0.35n + 0.3583333333 + n2 = 0.0 + 0.3583333333

Reorder the terms:
-0.3583333333 + 0.3583333333 + -0.35n + n2 = 0.0 + 0.3583333333

Combine like terms: -0.3583333333 + 0.3583333333 = 0.0000000000
0.0000000000 + -0.35n + n2 = 0.0 + 0.3583333333
-0.35n + n2 = 0.0 + 0.3583333333

Combine like terms: 0.0 + 0.3583333333 = 0.3583333333
-0.35n + n2 = 0.3583333333

The n term is -0.35n.  Take half its coefficient (-0.175).
Square it (0.030625) and add it to both sides.

Add '0.030625' to each side of the equation.
-0.35n + 0.030625 + n2 = 0.3583333333 + 0.030625

Reorder the terms:
0.030625 + -0.35n + n2 = 0.3583333333 + 0.030625

Combine like terms: 0.3583333333 + 0.030625 = 0.3889583333
0.030625 + -0.35n + n2 = 0.3889583333

Factor a perfect square on the left side:
(n + -0.175)(n + -0.175) = 0.3889583333

Calculate the square root of the right side: 0.623665241

Break this problem into two subproblems by setting 
(n + -0.175) equal to 0.623665241 and -0.623665241.

Subproblem 1
n + -0.175 = 0.623665241

Simplifying
n + -0.175 = 0.623665241

Reorder the terms:
-0.175 + n = 0.623665241

Solving
-0.175 + n = 0.623665241

Solving for variable 'n'.

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

Add '0.175' to each side of the equation.
-0.175 + 0.175 + n = 0.623665241 + 0.175

Combine like terms: -0.175 + 0.175 = 0.000
0.000 + n = 0.623665241 + 0.175
n = 0.623665241 + 0.175

Combine like terms: 0.623665241 + 0.175 = 0.798665241
n = 0.798665241

Simplifying
n = 0.798665241

Subproblem 2
n + -0.175 = -0.623665241

Simplifying
n + -0.175 = -0.623665241

Reorder the terms:
-0.175 + n = -0.623665241

Solving
-0.175 + n = -0.623665241

Solving for variable 'n'.

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

Add '0.175' to each side of the equation.
-0.175 + 0.175 + n = -0.623665241 + 0.175

Combine like terms: -0.175 + 0.175 = 0.000
0.000 + n = -0.623665241 + 0.175
n = -0.623665241 + 0.175

Combine like terms: -0.623665241 + 0.175 = -0.448665241
n = -0.448665241

Simplifying
n = -0.448665241

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.798665241, -0.448665241}