(u^2-1)=2u^2-u-5

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Solution for (u^2-1)=2u^2-u-5 equation: 


Simplifying
(u2 + -1) = 2u2 + -1u + -5

Reorder the terms:
(-1 + u2) = 2u2 + -1u + -5

Remove parenthesis around (-1 + u2)
-1 + u2 = 2u2 + -1u + -5

Reorder the terms:
-1 + u2 = -5 + -1u + 2u2

Solving
-1 + u2 = -5 + -1u + 2u2

Solving for variable 'u'.

Reorder the terms:
-1 + 5 + u + u2 + -2u2 = -5 + -1u + 2u2 + 5 + u + -2u2

Combine like terms: -1 + 5 = 4
4 + u + u2 + -2u2 = -5 + -1u + 2u2 + 5 + u + -2u2

Combine like terms: u2 + -2u2 = -1u2
4 + u + -1u2 = -5 + -1u + 2u2 + 5 + u + -2u2

Reorder the terms:
4 + u + -1u2 = -5 + 5 + -1u + u + 2u2 + -2u2

Combine like terms: -5 + 5 = 0
4 + u + -1u2 = 0 + -1u + u + 2u2 + -2u2
4 + u + -1u2 = -1u + u + 2u2 + -2u2

Combine like terms: -1u + u = 0
4 + u + -1u2 = 0 + 2u2 + -2u2
4 + u + -1u2 = 2u2 + -2u2

Combine like terms: 2u2 + -2u2 = 0
4 + u + -1u2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-4 + -1u + u2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + -1u + 4 + u2 = 0 + 4

Reorder the terms:
-4 + 4 + -1u + u2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + -1u + u2 = 0 + 4
-1u + u2 = 0 + 4

Combine like terms: 0 + 4 = 4
-1u + u2 = 4

The u term is u.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1u + 0.25 + u2 = 4 + 0.25

Reorder the terms:
0.25 + -1u + u2 = 4 + 0.25

Combine like terms: 4 + 0.25 = 4.25
0.25 + -1u + u2 = 4.25

Factor a perfect square on the left side:
(u + 0.5)(u + 0.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(u + 0.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
u + 0.5 = 2.061552813

Simplifying
u + 0.5 = 2.061552813

Reorder the terms:
0.5 + u = 2.061552813

Solving
0.5 + u = 2.061552813

Solving for variable 'u'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + u = 2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + u = 2.061552813 + -0.5
u = 2.061552813 + -0.5

Combine like terms: 2.061552813 + -0.5 = 1.561552813
u = 1.561552813

Simplifying
u = 1.561552813


Subproblem 2
u + 0.5 = -2.061552813

Simplifying
u + 0.5 = -2.061552813

Reorder the terms:
0.5 + u = -2.061552813

Solving
0.5 + u = -2.061552813

Solving for variable 'u'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + u = -2.061552813 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + u = -2.061552813 + -0.5
u = -2.061552813 + -0.5

Combine like terms: -2.061552813 + -0.5 = -2.561552813
u = -2.561552813

Simplifying
u = -2.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
u = {1.561552813, -2.561552813}

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