55-(2c+3)=4c(c+6)+c

Solution for 55-(2c+3)=4c(c+6)+c equation:

Simplifying
55 + -1(2c + 3) = 4c(c + 6) + c

Reorder the terms:
55 + -1(3 + 2c) = 4c(c + 6) + c
55 + (3 * -1 + 2c * -1) = 4c(c + 6) + c
55 + (-3 + -2c) = 4c(c + 6) + c

Combine like terms: 55 + -3 = 52
52 + -2c = 4c(c + 6) + c

Reorder the terms:
52 + -2c = 4c(6 + c) + c
52 + -2c = (6 * 4c + c * 4c) + c
52 + -2c = (24c + 4c2) + c

Reorder the terms:
52 + -2c = 24c + c + 4c2

Combine like terms: 24c + c = 25c
52 + -2c = 25c + 4c2

Solving
52 + -2c = 25c + 4c2

Solving for variable 'c'.

Combine like terms: -2c + -25c = -27c
52 + -27c + -4c2 = 25c + 4c2 + -25c + -4c2

Reorder the terms:
52 + -27c + -4c2 = 25c + -25c + 4c2 + -4c2

Combine like terms: 25c + -25c = 0
52 + -27c + -4c2 = 0 + 4c2 + -4c2
52 + -27c + -4c2 = 4c2 + -4c2

Combine like terms: 4c2 + -4c2 = 0
52 + -27c + -4c2 = 0

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

Divide each side by '-4'.
-13 + 6.75c + c2 = 0

Move the constant term to the right:

Add '13' to each side of the equation.
-13 + 6.75c + 13 + c2 = 0 + 13

Reorder the terms:
-13 + 13 + 6.75c + c2 = 0 + 13

Combine like terms: -13 + 13 = 0
0 + 6.75c + c2 = 0 + 13
6.75c + c2 = 0 + 13

Combine like terms: 0 + 13 = 13
6.75c + c2 = 13

The c term is 6.75c.  Take half its coefficient (3.375).
Square it (11.390625) and add it to both sides.

Add '11.390625' to each side of the equation.
6.75c + 11.390625 + c2 = 13 + 11.390625

Reorder the terms:
11.390625 + 6.75c + c2 = 13 + 11.390625

Combine like terms: 13 + 11.390625 = 24.390625
11.390625 + 6.75c + c2 = 24.390625

Factor a perfect square on the left side:
(c + 3.375)(c + 3.375) = 24.390625

Calculate the square root of the right side: 4.938686566

Break this problem into two subproblems by setting 
(c + 3.375) equal to 4.938686566 and -4.938686566.

Subproblem 1
c + 3.375 = 4.938686566

Simplifying
c + 3.375 = 4.938686566

Reorder the terms:
3.375 + c = 4.938686566

Solving
3.375 + c = 4.938686566

Solving for variable 'c'.

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

Add '-3.375' to each side of the equation.
3.375 + -3.375 + c = 4.938686566 + -3.375

Combine like terms: 3.375 + -3.375 = 0.000
0.000 + c = 4.938686566 + -3.375
c = 4.938686566 + -3.375

Combine like terms: 4.938686566 + -3.375 = 1.563686566
c = 1.563686566

Simplifying
c = 1.563686566

Subproblem 2
c + 3.375 = -4.938686566

Simplifying
c + 3.375 = -4.938686566

Reorder the terms:
3.375 + c = -4.938686566

Solving
3.375 + c = -4.938686566

Solving for variable 'c'.

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

Add '-3.375' to each side of the equation.
3.375 + -3.375 + c = -4.938686566 + -3.375

Combine like terms: 3.375 + -3.375 = 0.000
0.000 + c = -4.938686566 + -3.375
c = -4.938686566 + -3.375

Combine like terms: -4.938686566 + -3.375 = -8.313686566
c = -8.313686566

Simplifying
c = -8.313686566

Solution
The solution to the problem is based on the solutions
from the subproblems.
c = {1.563686566, -8.313686566}