n^2(n^2-5)=2n^2-3

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Solution for n^2(n^2-5)=2n^2-3 equation: 


Simplifying
n2(n2 + -5) = 2n2 + -3

Reorder the terms:
n2(-5 + n2) = 2n2 + -3
(-5 * n2 + n2 * n2) = 2n2 + -3
(-5n2 + n4) = 2n2 + -3

Reorder the terms:
-5n2 + n4 = -3 + 2n2

Solving
-5n2 + n4 = -3 + 2n2

Solving for variable 'n'.

Reorder the terms:
3 + -5n2 + -2n2 + n4 = -3 + 2n2 + 3 + -2n2

Combine like terms: -5n2 + -2n2 = -7n2
3 + -7n2 + n4 = -3 + 2n2 + 3 + -2n2

Reorder the terms:
3 + -7n2 + n4 = -3 + 3 + 2n2 + -2n2

Combine like terms: -3 + 3 = 0
3 + -7n2 + n4 = 0 + 2n2 + -2n2
3 + -7n2 + n4 = 2n2 + -2n2

Combine like terms: 2n2 + -2n2 = 0
3 + -7n2 + n4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -7n2 + -3 + n4 = 0 + -3

Reorder the terms:
3 + -3 + -7n2 + n4 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -7n2 + n4 = 0 + -3
-7n2 + n4 = 0 + -3

Combine like terms: 0 + -3 = -3
-7n2 + n4 = -3

The n term is -7n2.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7n2 + 12.25 + n4 = -3 + 12.25

Reorder the terms:
12.25 + -7n2 + n4 = -3 + 12.25

Combine like terms: -3 + 12.25 = 9.25
12.25 + -7n2 + n4 = 9.25

Factor a perfect square on the left side:
(n2 + -3.5)(n2 + -3.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(n2 + -3.5) equal to 3.041381265 and -3.041381265.


Subproblem 1
n2 + -3.5 = 3.041381265

Simplifying
n2 + -3.5 = 3.041381265

Reorder the terms:
-3.5 + n2 = 3.041381265

Solving
-3.5 + n2 = 3.041381265

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n2 = 3.041381265 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n2 = 3.041381265 + 3.5
n2 = 3.041381265 + 3.5

Combine like terms: 3.041381265 + 3.5 = 6.541381265
n2 = 6.541381265

Simplifying
n2 = 6.541381265

Take the square root of each side:
n = {-2.557612415, 2.557612415}


Subproblem 2
n2 + -3.5 = -3.041381265

Simplifying
n2 + -3.5 = -3.041381265

Reorder the terms:
-3.5 + n2 = -3.041381265

Solving
-3.5 + n2 = -3.041381265

Solving for variable 'n'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + n2 = -3.041381265 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + n2 = -3.041381265 + 3.5
n2 = -3.041381265 + 3.5

Combine like terms: -3.041381265 + 3.5 = 0.458618735
n2 = 0.458618735

Simplifying
n2 = 0.458618735

Take the square root of each side:
n = {-0.677213951, 0.677213951}


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-2.557612415, 2.557612415, -0.677213951, 0.677213951}

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