41u=u^2-5

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Solution for 41u=u^2-5 equation:


Simplifying
41u = u2 + -5

Reorder the terms:
41u = -5 + u2

Solving
41u = -5 + u2

Solving for variable 'u'.

Reorder the terms:
5 + 41u + -1u2 = -5 + u2 + 5 + -1u2

Reorder the terms:
5 + 41u + -1u2 = -5 + 5 + u2 + -1u2

Combine like terms: -5 + 5 = 0
5 + 41u + -1u2 = 0 + u2 + -1u2
5 + 41u + -1u2 = u2 + -1u2

Combine like terms: u2 + -1u2 = 0
5 + 41u + -1u2 = 0

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

Divide each side by '-1'.
-5 + -41u + u2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -41u + 5 + u2 = 0 + 5

Reorder the terms:
-5 + 5 + -41u + u2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -41u + u2 = 0 + 5
-41u + u2 = 0 + 5

Combine like terms: 0 + 5 = 5
-41u + u2 = 5

The u term is -41u.  Take half its coefficient (-20.5).
Square it (420.25) and add it to both sides.

Add '420.25' to each side of the equation.
-41u + 420.25 + u2 = 5 + 420.25

Reorder the terms:
420.25 + -41u + u2 = 5 + 420.25

Combine like terms: 5 + 420.25 = 425.25
420.25 + -41u + u2 = 425.25

Factor a perfect square on the left side:
(u + -20.5)(u + -20.5) = 425.25

Calculate the square root of the right side: 20.621590627

Break this problem into two subproblems by setting 
(u + -20.5) equal to 20.621590627 and -20.621590627.

Subproblem 1

u + -20.5 = 20.621590627 Simplifying u + -20.5 = 20.621590627 Reorder the terms: -20.5 + u = 20.621590627 Solving -20.5 + u = 20.621590627 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '20.5' to each side of the equation. -20.5 + 20.5 + u = 20.621590627 + 20.5 Combine like terms: -20.5 + 20.5 = 0.0 0.0 + u = 20.621590627 + 20.5 u = 20.621590627 + 20.5 Combine like terms: 20.621590627 + 20.5 = 41.121590627 u = 41.121590627 Simplifying u = 41.121590627

Subproblem 2

u + -20.5 = -20.621590627 Simplifying u + -20.5 = -20.621590627 Reorder the terms: -20.5 + u = -20.621590627 Solving -20.5 + u = -20.621590627 Solving for variable 'u'. Move all terms containing u to the left, all other terms to the right. Add '20.5' to each side of the equation. -20.5 + 20.5 + u = -20.621590627 + 20.5 Combine like terms: -20.5 + 20.5 = 0.0 0.0 + u = -20.621590627 + 20.5 u = -20.621590627 + 20.5 Combine like terms: -20.621590627 + 20.5 = -0.121590627 u = -0.121590627 Simplifying u = -0.121590627

Solution

The solution to the problem is based on the solutions from the subproblems. u = {41.121590627, -0.121590627}

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