-4u^2+16=9u-5u^2

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Solution for -4u^2+16=9u-5u^2 equation: 


Simplifying
-4u2 + 16 = 9u + -5u2

Reorder the terms:
16 + -4u2 = 9u + -5u2

Solving
16 + -4u2 = 9u + -5u2

Solving for variable 'u'.

Reorder the terms:
16 + -9u + -4u2 + 5u2 = 9u + -5u2 + -9u + 5u2

Combine like terms: -4u2 + 5u2 = 1u2
16 + -9u + 1u2 = 9u + -5u2 + -9u + 5u2

Reorder the terms:
16 + -9u + 1u2 = 9u + -9u + -5u2 + 5u2

Combine like terms: 9u + -9u = 0
16 + -9u + 1u2 = 0 + -5u2 + 5u2
16 + -9u + 1u2 = -5u2 + 5u2

Combine like terms: -5u2 + 5u2 = 0
16 + -9u + 1u2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-16' to each side of the equation.
16 + -9u + -16 + u2 = 0 + -16

Reorder the terms:
16 + -16 + -9u + u2 = 0 + -16

Combine like terms: 16 + -16 = 0
0 + -9u + u2 = 0 + -16
-9u + u2 = 0 + -16

Combine like terms: 0 + -16 = -16
-9u + u2 = -16

The u term is -9u.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9u + 20.25 + u2 = -16 + 20.25

Reorder the terms:
20.25 + -9u + u2 = -16 + 20.25

Combine like terms: -16 + 20.25 = 4.25
20.25 + -9u + u2 = 4.25

Factor a perfect square on the left side:
(u + -4.5)(u + -4.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(u + -4.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
u + -4.5 = 2.061552813

Simplifying
u + -4.5 = 2.061552813

Reorder the terms:
-4.5 + u = 2.061552813

Solving
-4.5 + u = 2.061552813

Solving for variable 'u'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + u = 2.061552813 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + u = 2.061552813 + 4.5
u = 2.061552813 + 4.5

Combine like terms: 2.061552813 + 4.5 = 6.561552813
u = 6.561552813

Simplifying
u = 6.561552813


Subproblem 2
u + -4.5 = -2.061552813

Simplifying
u + -4.5 = -2.061552813

Reorder the terms:
-4.5 + u = -2.061552813

Solving
-4.5 + u = -2.061552813

Solving for variable 'u'.

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

Add '4.5' to each side of the equation.
-4.5 + 4.5 + u = -2.061552813 + 4.5

Combine like terms: -4.5 + 4.5 = 0.0
0.0 + u = -2.061552813 + 4.5
u = -2.061552813 + 4.5

Combine like terms: -2.061552813 + 4.5 = 2.438447187
u = 2.438447187

Simplifying
u = 2.438447187


Solution
The solution to the problem is based on the solutions
from the subproblems.
u = {6.561552813, 2.438447187}

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