25(3n+2)(n-1)=252900

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Solution for 25(3n+2)(n-1)=252900 equation: 


Simplifying
25(3n + 2)(n + -1) = 252900

Reorder the terms:
25(2 + 3n)(n + -1) = 252900

Reorder the terms:
25(2 + 3n)(-1 + n) = 252900

Multiply (2 + 3n) * (-1 + n)
25(2(-1 + n) + 3n * (-1 + n)) = 252900
25((-1 * 2 + n * 2) + 3n * (-1 + n)) = 252900
25((-2 + 2n) + 3n * (-1 + n)) = 252900
25(-2 + 2n + (-1 * 3n + n * 3n)) = 252900
25(-2 + 2n + (-3n + 3n2)) = 252900

Combine like terms: 2n + -3n = -1n
25(-2 + -1n + 3n2) = 252900
(-2 * 25 + -1n * 25 + 3n2 * 25) = 252900
(-50 + -25n + 75n2) = 252900

Solving
-50 + -25n + 75n2 = 252900

Solving for variable 'n'.

Reorder the terms:
-50 + -252900 + -25n + 75n2 = 252900 + -252900

Combine like terms: -50 + -252900 = -252950
-252950 + -25n + 75n2 = 252900 + -252900

Combine like terms: 252900 + -252900 = 0
-252950 + -25n + 75n2 = 0

Factor out the Greatest Common Factor (GCF), '25'.
25(-10118 + -1n + 3n2) = 0

Ignore the factor 25.

Subproblem 1
Set the factor '(-10118 + -1n + 3n2)' equal to zero and attempt to solve:

Simplifying
-10118 + -1n + 3n2 = 0

Solving
-10118 + -1n + 3n2 = 0

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

Divide each side by '3'.
-3372.666667 + -0.3333333333n + n2 = 0

Move the constant term to the right:

Add '3372.666667' to each side of the equation.
-3372.666667 + -0.3333333333n + 3372.666667 + n2 = 0 + 3372.666667

Reorder the terms:
-3372.666667 + 3372.666667 + -0.3333333333n + n2 = 0 + 3372.666667

Combine like terms: -3372.666667 + 3372.666667 = 0.000000
0.000000 + -0.3333333333n + n2 = 0 + 3372.666667
-0.3333333333n + n2 = 0 + 3372.666667

Combine like terms: 0 + 3372.666667 = 3372.666667
-0.3333333333n + n2 = 3372.666667

The n term is -0.3333333333n.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333n + 0.02777777779 + n2 = 3372.666667 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333n + n2 = 3372.666667 + 0.02777777779

Combine like terms: 3372.666667 + 0.02777777779 = 3372.69444477779
0.02777777779 + -0.3333333333n + n2 = 3372.69444477779

Factor a perfect square on the left side:
(n + -0.1666666667)(n + -0.1666666667) = 3372.69444477779

Calculate the square root of the right side: 58.074903743

Break this problem into two subproblems by setting 
(n + -0.1666666667) equal to 58.074903743 and -58.074903743.


Subproblem 1
n + -0.1666666667 = 58.074903743

Simplifying
n + -0.1666666667 = 58.074903743

Reorder the terms:
-0.1666666667 + n = 58.074903743

Solving
-0.1666666667 + n = 58.074903743

Solving for variable 'n'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + n = 58.074903743 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + n = 58.074903743 + 0.1666666667
n = 58.074903743 + 0.1666666667

Combine like terms: 58.074903743 + 0.1666666667 = 58.2415704097
n = 58.2415704097

Simplifying
n = 58.2415704097


Subproblem 2
n + -0.1666666667 = -58.074903743

Simplifying
n + -0.1666666667 = -58.074903743

Reorder the terms:
-0.1666666667 + n = -58.074903743

Solving
-0.1666666667 + n = -58.074903743

Solving for variable 'n'.

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

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 0.1666666667 + n = -58.074903743 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + n = -58.074903743 + 0.1666666667
n = -58.074903743 + 0.1666666667

Combine like terms: -58.074903743 + 0.1666666667 = -57.9082370763
n = -57.9082370763

Simplifying
n = -57.9082370763


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {58.2415704097, -57.9082370763}
Solution
n = {58.2415704097, -57.9082370763}

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