c(c)+3c-54=c

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Solution for c(c)+3c-54=c equation:


Simplifying
c(c) + 3c + -54 = c

Multiply c * c
c2 + 3c + -54 = c

Reorder the terms:
-54 + 3c + c2 = c

Solving
-54 + 3c + c2 = c

Solving for variable 'c'.

Reorder the terms:
-54 + 3c + -1c + c2 = c + -1c

Combine like terms: 3c + -1c = 2c
-54 + 2c + c2 = c + -1c

Combine like terms: c + -1c = 0
-54 + 2c + c2 = 0

Begin completing the square.

Move the constant term to the right:

Add '54' to each side of the equation.
-54 + 2c + 54 + c2 = 0 + 54

Reorder the terms:
-54 + 54 + 2c + c2 = 0 + 54

Combine like terms: -54 + 54 = 0
0 + 2c + c2 = 0 + 54
2c + c2 = 0 + 54

Combine like terms: 0 + 54 = 54
2c + c2 = 54

The c term is 2c.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2c + 1 + c2 = 54 + 1

Reorder the terms:
1 + 2c + c2 = 54 + 1

Combine like terms: 54 + 1 = 55
1 + 2c + c2 = 55

Factor a perfect square on the left side:
(c + 1)(c + 1) = 55

Calculate the square root of the right side: 7.416198487

Break this problem into two subproblems by setting 
(c + 1) equal to 7.416198487 and -7.416198487.

Subproblem 1

c + 1 = 7.416198487 Simplifying c + 1 = 7.416198487 Reorder the terms: 1 + c = 7.416198487 Solving 1 + c = 7.416198487 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + c = 7.416198487 + -1 Combine like terms: 1 + -1 = 0 0 + c = 7.416198487 + -1 c = 7.416198487 + -1 Combine like terms: 7.416198487 + -1 = 6.416198487 c = 6.416198487 Simplifying c = 6.416198487

Subproblem 2

c + 1 = -7.416198487 Simplifying c + 1 = -7.416198487 Reorder the terms: 1 + c = -7.416198487 Solving 1 + c = -7.416198487 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + c = -7.416198487 + -1 Combine like terms: 1 + -1 = 0 0 + c = -7.416198487 + -1 c = -7.416198487 + -1 Combine like terms: -7.416198487 + -1 = -8.416198487 c = -8.416198487 Simplifying c = -8.416198487

Solution

The solution to the problem is based on the solutions from the subproblems. c = {6.416198487, -8.416198487}

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