5.5x-1-4.5x^2=0

Solution for 5.5x-1-4.5x^2=0 equation:

Simplifying
5.5x + -1 + -4.5x2 = 0

Reorder the terms:
-1 + 5.5x + -4.5x2 = 0

Solving
-1 + 5.5x + -4.5x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-4.5 the coefficient of the squared term: 

Divide each side by '-4.5'.
0.2222222222 + -1.222222222x + x2 = 0

Move the constant term to the right:

Add '-0.2222222222' to each side of the equation.
0.2222222222 + -1.222222222x + -0.2222222222 + x2 = 0 + -0.2222222222

Reorder the terms:
0.2222222222 + -0.2222222222 + -1.222222222x + x2 = 0 + -0.2222222222

Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000
0.0000000000 + -1.222222222x + x2 = 0 + -0.2222222222
-1.222222222x + x2 = 0 + -0.2222222222

Combine like terms: 0 + -0.2222222222 = -0.2222222222
-1.222222222x + x2 = -0.2222222222

The x term is -1.222222222x.  Take half its coefficient (-0.611111111).
Square it (0.3734567900) and add it to both sides.

Add '0.3734567900' to each side of the equation.
-1.222222222x + 0.3734567900 + x2 = -0.2222222222 + 0.3734567900

Reorder the terms:
0.3734567900 + -1.222222222x + x2 = -0.2222222222 + 0.3734567900

Combine like terms: -0.2222222222 + 0.3734567900 = 0.1512345678
0.3734567900 + -1.222222222x + x2 = 0.1512345678

Factor a perfect square on the left side:
(x + -0.611111111)(x + -0.611111111) = 0.1512345678

Calculate the square root of the right side: 0.388888889

Break this problem into two subproblems by setting 
(x + -0.611111111) equal to 0.388888889 and -0.388888889.

Subproblem 1
x + -0.611111111 = 0.388888889

Simplifying
x + -0.611111111 = 0.388888889

Reorder the terms:
-0.611111111 + x = 0.388888889

Solving
-0.611111111 + x = 0.388888889

Solving for variable 'x'.

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

Add '0.611111111' to each side of the equation.
-0.611111111 + 0.611111111 + x = 0.388888889 + 0.611111111

Combine like terms: -0.611111111 + 0.611111111 = 0.000000000
0.000000000 + x = 0.388888889 + 0.611111111
x = 0.388888889 + 0.611111111

Combine like terms: 0.388888889 + 0.611111111 = 1
x = 1

Simplifying
x = 1

Subproblem 2
x + -0.611111111 = -0.388888889

Simplifying
x + -0.611111111 = -0.388888889

Reorder the terms:
-0.611111111 + x = -0.388888889

Solving
-0.611111111 + x = -0.388888889

Solving for variable 'x'.

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

Add '0.611111111' to each side of the equation.
-0.611111111 + 0.611111111 + x = -0.388888889 + 0.611111111

Combine like terms: -0.611111111 + 0.611111111 = 0.000000000
0.000000000 + x = -0.388888889 + 0.611111111
x = -0.388888889 + 0.611111111

Combine like terms: -0.388888889 + 0.611111111 = 0.222222222
x = 0.222222222

Simplifying
x = 0.222222222

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1, 0.222222222}