(2n^4+10n^2)-(4n^4-5n^2+5)=

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Solution for (2n^4+10n^2)-(4n^4-5n^2+5)= equation: 


Simplifying
(2n4 + 10n2) + -1(4n4 + -5n2 + 5) = 0

Reorder the terms:
(10n2 + 2n4) + -1(4n4 + -5n2 + 5) = 0

Remove parenthesis around (10n2 + 2n4)
10n2 + 2n4 + -1(4n4 + -5n2 + 5) = 0

Reorder the terms:
10n2 + 2n4 + -1(5 + -5n2 + 4n4) = 0
10n2 + 2n4 + (5 * -1 + -5n2 * -1 + 4n4 * -1) = 0
10n2 + 2n4 + (-5 + 5n2 + -4n4) = 0

Reorder the terms:
-5 + 10n2 + 5n2 + 2n4 + -4n4 = 0

Combine like terms: 10n2 + 5n2 = 15n2
-5 + 15n2 + 2n4 + -4n4 = 0

Combine like terms: 2n4 + -4n4 = -2n4
-5 + 15n2 + -2n4 = 0

Solving
-5 + 15n2 + -2n4 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
2.5 + -7.5n2 + n4 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + -7.5n2 + -2.5 + n4 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + -7.5n2 + n4 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + -7.5n2 + n4 = 0 + -2.5
-7.5n2 + n4 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
-7.5n2 + n4 = -2.5

The n term is -7.5n2.  Take half its coefficient (-3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
-7.5n2 + 14.0625 + n4 = -2.5 + 14.0625

Reorder the terms:
14.0625 + -7.5n2 + n4 = -2.5 + 14.0625

Combine like terms: -2.5 + 14.0625 = 11.5625
14.0625 + -7.5n2 + n4 = 11.5625

Factor a perfect square on the left side:
(n2 + -3.75)(n2 + -3.75) = 11.5625

Calculate the square root of the right side: 3.400367627

Break this problem into two subproblems by setting 
(n2 + -3.75) equal to 3.400367627 and -3.400367627.


Subproblem 1
n2 + -3.75 = 3.400367627

Simplifying
n2 + -3.75 = 3.400367627

Reorder the terms:
-3.75 + n2 = 3.400367627

Solving
-3.75 + n2 = 3.400367627

Solving for variable 'n'.

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

Add '3.75' to each side of the equation.
-3.75 + 3.75 + n2 = 3.400367627 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + n2 = 3.400367627 + 3.75
n2 = 3.400367627 + 3.75

Combine like terms: 3.400367627 + 3.75 = 7.150367627
n2 = 7.150367627

Simplifying
n2 = 7.150367627

Take the square root of each side:
n = {-2.674017133, 2.674017133}


Subproblem 2
n2 + -3.75 = -3.400367627

Simplifying
n2 + -3.75 = -3.400367627

Reorder the terms:
-3.75 + n2 = -3.400367627

Solving
-3.75 + n2 = -3.400367627

Solving for variable 'n'.

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

Add '3.75' to each side of the equation.
-3.75 + 3.75 + n2 = -3.400367627 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + n2 = -3.400367627 + 3.75
n2 = -3.400367627 + 3.75

Combine like terms: -3.400367627 + 3.75 = 0.349632373
n2 = 0.349632373

Simplifying
n2 = 0.349632373

Take the square root of each side:
n = {-0.591297195, 0.591297195}


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-2.674017133, 2.674017133, -0.591297195, 0.591297195}

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