6x-54=-3x(x+45)

Solution for 6x-54=-3x(x+45) equation:

Simplifying
6x + -54 = -3x(x + 45)

Reorder the terms:
-54 + 6x = -3x(x + 45)

Reorder the terms:
-54 + 6x = -3x(45 + x)
-54 + 6x = (45 * -3x + x * -3x)
-54 + 6x = (-135x + -3x2)

Solving
-54 + 6x = -135x + -3x2

Solving for variable 'x'.

Combine like terms: 6x + 135x = 141x
-54 + 141x + 3x2 = -135x + -3x2 + 135x + 3x2

Reorder the terms:
-54 + 141x + 3x2 = -135x + 135x + -3x2 + 3x2

Combine like terms: -135x + 135x = 0
-54 + 141x + 3x2 = 0 + -3x2 + 3x2
-54 + 141x + 3x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-54 + 141x + 3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-18 + 47x + x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-18 + 47x + x2)' equal to zero and attempt to solve:

Simplifying
-18 + 47x + x2 = 0

Solving
-18 + 47x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '18' to each side of the equation.
-18 + 47x + 18 + x2 = 0 + 18

Reorder the terms:
-18 + 18 + 47x + x2 = 0 + 18

Combine like terms: -18 + 18 = 0
0 + 47x + x2 = 0 + 18
47x + x2 = 0 + 18

Combine like terms: 0 + 18 = 18
47x + x2 = 18

The x term is 47x.  Take half its coefficient (23.5).
Square it (552.25) and add it to both sides.

Add '552.25' to each side of the equation.
47x + 552.25 + x2 = 18 + 552.25

Reorder the terms:
552.25 + 47x + x2 = 18 + 552.25

Combine like terms: 18 + 552.25 = 570.25
552.25 + 47x + x2 = 570.25

Factor a perfect square on the left side:
(x + 23.5)(x + 23.5) = 570.25

Calculate the square root of the right side: 23.879907873

Break this problem into two subproblems by setting 
(x + 23.5) equal to 23.879907873 and -23.879907873.

Subproblem 1
x + 23.5 = 23.879907873

Simplifying
x + 23.5 = 23.879907873

Reorder the terms:
23.5 + x = 23.879907873

Solving
23.5 + x = 23.879907873

Solving for variable 'x'.

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

Add '-23.5' to each side of the equation.
23.5 + -23.5 + x = 23.879907873 + -23.5

Combine like terms: 23.5 + -23.5 = 0.0
0.0 + x = 23.879907873 + -23.5
x = 23.879907873 + -23.5

Combine like terms: 23.879907873 + -23.5 = 0.379907873
x = 0.379907873

Simplifying
x = 0.379907873

Subproblem 2
x + 23.5 = -23.879907873

Simplifying
x + 23.5 = -23.879907873

Reorder the terms:
23.5 + x = -23.879907873

Solving
23.5 + x = -23.879907873

Solving for variable 'x'.

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

Add '-23.5' to each side of the equation.
23.5 + -23.5 + x = -23.879907873 + -23.5

Combine like terms: 23.5 + -23.5 = 0.0
0.0 + x = -23.879907873 + -23.5
x = -23.879907873 + -23.5

Combine like terms: -23.879907873 + -23.5 = -47.379907873
x = -47.379907873

Simplifying
x = -47.379907873

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.379907873, -47.379907873}
Solution
x = {0.379907873, -47.379907873}