9n^2-30n=25

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Solution for 9n^2-30n=25 equation: 


Simplifying
9n2 + -30n = 25

Reorder the terms:
-30n + 9n2 = 25

Solving
-30n + 9n2 = 25

Solving for variable 'n'.

Reorder the terms:
-25 + -30n + 9n2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + -30n + 9n2 = 0

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

Divide each side by '9'.
-2.777777778 + -3.333333333n + n2 = 0

Move the constant term to the right:

Add '2.777777778' to each side of the equation.
-2.777777778 + -3.333333333n + 2.777777778 + n2 = 0 + 2.777777778

Reorder the terms:
-2.777777778 + 2.777777778 + -3.333333333n + n2 = 0 + 2.777777778

Combine like terms: -2.777777778 + 2.777777778 = 0.000000000
0.000000000 + -3.333333333n + n2 = 0 + 2.777777778
-3.333333333n + n2 = 0 + 2.777777778

Combine like terms: 0 + 2.777777778 = 2.777777778
-3.333333333n + n2 = 2.777777778

The n term is -3.333333333n.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333n + 2.777777779 + n2 = 2.777777778 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333n + n2 = 2.777777778 + 2.777777779

Combine like terms: 2.777777778 + 2.777777779 = 5.555555557
2.777777779 + -3.333333333n + n2 = 5.555555557

Factor a perfect square on the left side:
(n + -1.666666667)(n + -1.666666667) = 5.555555557

Calculate the square root of the right side: 2.357022604

Break this problem into two subproblems by setting 
(n + -1.666666667) equal to 2.357022604 and -2.357022604.


Subproblem 1
n + -1.666666667 = 2.357022604

Simplifying
n + -1.666666667 = 2.357022604

Reorder the terms:
-1.666666667 + n = 2.357022604

Solving
-1.666666667 + n = 2.357022604

Solving for variable 'n'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + n = 2.357022604 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + n = 2.357022604 + 1.666666667
n = 2.357022604 + 1.666666667

Combine like terms: 2.357022604 + 1.666666667 = 4.023689271
n = 4.023689271

Simplifying
n = 4.023689271


Subproblem 2
n + -1.666666667 = -2.357022604

Simplifying
n + -1.666666667 = -2.357022604

Reorder the terms:
-1.666666667 + n = -2.357022604

Solving
-1.666666667 + n = -2.357022604

Solving for variable 'n'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + n = -2.357022604 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + n = -2.357022604 + 1.666666667
n = -2.357022604 + 1.666666667

Combine like terms: -2.357022604 + 1.666666667 = -0.690355937
n = -0.690355937

Simplifying
n = -0.690355937


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.023689271, -0.690355937}

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