4x^2+33x+54.55=0

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Solution for 4x^2+33x+54.55=0 equation: 


Simplifying
4x2 + 33x + 54.55 = 0

Reorder the terms:
54.55 + 33x + 4x2 = 0

Solving
54.55 + 33x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
13.6375 + 8.25x + x2 = 0

Move the constant term to the right:

Add '-13.6375' to each side of the equation.
13.6375 + 8.25x + -13.6375 + x2 = 0 + -13.6375

Reorder the terms:
13.6375 + -13.6375 + 8.25x + x2 = 0 + -13.6375

Combine like terms: 13.6375 + -13.6375 = 0.0000
0.0000 + 8.25x + x2 = 0 + -13.6375
8.25x + x2 = 0 + -13.6375

Combine like terms: 0 + -13.6375 = -13.6375
8.25x + x2 = -13.6375

The x term is 8.25x.  Take half its coefficient (4.125).
Square it (17.015625) and add it to both sides.

Add '17.015625' to each side of the equation.
8.25x + 17.015625 + x2 = -13.6375 + 17.015625

Reorder the terms:
17.015625 + 8.25x + x2 = -13.6375 + 17.015625

Combine like terms: -13.6375 + 17.015625 = 3.378125
17.015625 + 8.25x + x2 = 3.378125

Factor a perfect square on the left side:
(x + 4.125)(x + 4.125) = 3.378125

Calculate the square root of the right side: 1.837967628

Break this problem into two subproblems by setting 
(x + 4.125) equal to 1.837967628 and -1.837967628.


Subproblem 1
x + 4.125 = 1.837967628

Simplifying
x + 4.125 = 1.837967628

Reorder the terms:
4.125 + x = 1.837967628

Solving
4.125 + x = 1.837967628

Solving for variable 'x'.

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

Add '-4.125' to each side of the equation.
4.125 + -4.125 + x = 1.837967628 + -4.125

Combine like terms: 4.125 + -4.125 = 0.000
0.000 + x = 1.837967628 + -4.125
x = 1.837967628 + -4.125

Combine like terms: 1.837967628 + -4.125 = -2.287032372
x = -2.287032372

Simplifying
x = -2.287032372


Subproblem 2
x + 4.125 = -1.837967628

Simplifying
x + 4.125 = -1.837967628

Reorder the terms:
4.125 + x = -1.837967628

Solving
4.125 + x = -1.837967628

Solving for variable 'x'.

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

Add '-4.125' to each side of the equation.
4.125 + -4.125 + x = -1.837967628 + -4.125

Combine like terms: 4.125 + -4.125 = 0.000
0.000 + x = -1.837967628 + -4.125
x = -1.837967628 + -4.125

Combine like terms: -1.837967628 + -4.125 = -5.962967628
x = -5.962967628

Simplifying
x = -5.962967628


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.287032372, -5.962967628}

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