T^4-5T^2+5=0

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Solution for T^4-5T^2+5=0 equation: 


Simplifying
T4 + -5T2 + 5 = 0

Reorder the terms:
5 + -5T2 + T4 = 0

Solving
5 + -5T2 + T4 = 0

Solving for variable 'T'.

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -5T2 + -5 + T4 = 0 + -5

Reorder the terms:
5 + -5 + -5T2 + T4 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -5T2 + T4 = 0 + -5
-5T2 + T4 = 0 + -5

Combine like terms: 0 + -5 = -5
-5T2 + T4 = -5

The T term is -5T2.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5T2 + 6.25 + T4 = -5 + 6.25

Reorder the terms:
6.25 + -5T2 + T4 = -5 + 6.25

Combine like terms: -5 + 6.25 = 1.25
6.25 + -5T2 + T4 = 1.25

Factor a perfect square on the left side:
(T2 + -2.5)(T2 + -2.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(T2 + -2.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
T2 + -2.5 = 1.118033989

Simplifying
T2 + -2.5 = 1.118033989

Reorder the terms:
-2.5 + T2 = 1.118033989

Solving
-2.5 + T2 = 1.118033989

Solving for variable 'T'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + T2 = 1.118033989 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + T2 = 1.118033989 + 2.5
T2 = 1.118033989 + 2.5

Combine like terms: 1.118033989 + 2.5 = 3.618033989
T2 = 3.618033989

Simplifying
T2 = 3.618033989

Take the square root of each side:
T = {-1.902113033, 1.902113033}


Subproblem 2
T2 + -2.5 = -1.118033989

Simplifying
T2 + -2.5 = -1.118033989

Reorder the terms:
-2.5 + T2 = -1.118033989

Solving
-2.5 + T2 = -1.118033989

Solving for variable 'T'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + T2 = -1.118033989 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + T2 = -1.118033989 + 2.5
T2 = -1.118033989 + 2.5

Combine like terms: -1.118033989 + 2.5 = 1.381966011
T2 = 1.381966011

Simplifying
T2 = 1.381966011

Take the square root of each side:
T = {-1.175570504, 1.175570504}


Solution
The solution to the problem is based on the solutions
from the subproblems.
T = {-1.902113033, 1.902113033, -1.175570504, 1.175570504}

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