51(x+3)9x-23=20(4x+8)

Solution for 51(x+3)9x-23=20(4x+8) equation:

Simplifying
51(x + 3) * 9x + -23 = 20(4x + 8)

Reorder the terms:
51(3 + x) * 9x + -23 = 20(4x + 8)

Reorder the terms for easier multiplication:
51 * 9x(3 + x) + -23 = 20(4x + 8)

Multiply 51 * 9
459x(3 + x) + -23 = 20(4x + 8)
(3 * 459x + x * 459x) + -23 = 20(4x + 8)
(1377x + 459x2) + -23 = 20(4x + 8)

Reorder the terms:
-23 + 1377x + 459x2 = 20(4x + 8)

Reorder the terms:
-23 + 1377x + 459x2 = 20(8 + 4x)
-23 + 1377x + 459x2 = (8 * 20 + 4x * 20)
-23 + 1377x + 459x2 = (160 + 80x)

Solving
-23 + 1377x + 459x2 = 160 + 80x

Solving for variable 'x'.

Reorder the terms:
-23 + -160 + 1377x + -80x + 459x2 = 160 + 80x + -160 + -80x

Combine like terms: -23 + -160 = -183
-183 + 1377x + -80x + 459x2 = 160 + 80x + -160 + -80x

Combine like terms: 1377x + -80x = 1297x
-183 + 1297x + 459x2 = 160 + 80x + -160 + -80x

Reorder the terms:
-183 + 1297x + 459x2 = 160 + -160 + 80x + -80x

Combine like terms: 160 + -160 = 0
-183 + 1297x + 459x2 = 0 + 80x + -80x
-183 + 1297x + 459x2 = 80x + -80x

Combine like terms: 80x + -80x = 0
-183 + 1297x + 459x2 = 0

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

Divide each side by '459'.
-0.3986928105 + 2.825708061x + x2 = 0

Move the constant term to the right:

Add '0.3986928105' to each side of the equation.
-0.3986928105 + 2.825708061x + 0.3986928105 + x2 = 0 + 0.3986928105

Reorder the terms:
-0.3986928105 + 0.3986928105 + 2.825708061x + x2 = 0 + 0.3986928105

Combine like terms: -0.3986928105 + 0.3986928105 = 0.0000000000
0.0000000000 + 2.825708061x + x2 = 0 + 0.3986928105
2.825708061x + x2 = 0 + 0.3986928105

Combine like terms: 0 + 0.3986928105 = 0.3986928105
2.825708061x + x2 = 0.3986928105

The x term is 2.825708061x.  Take half its coefficient (1.412854031).
Square it (1.996156513) and add it to both sides.

Add '1.996156513' to each side of the equation.
2.825708061x + 1.996156513 + x2 = 0.3986928105 + 1.996156513

Reorder the terms:
1.996156513 + 2.825708061x + x2 = 0.3986928105 + 1.996156513

Combine like terms: 0.3986928105 + 1.996156513 = 2.3948493235
1.996156513 + 2.825708061x + x2 = 2.3948493235

Factor a perfect square on the left side:
(x + 1.412854031)(x + 1.412854031) = 2.3948493235

Calculate the square root of the right side: 1.547530072

Break this problem into two subproblems by setting 
(x + 1.412854031) equal to 1.547530072 and -1.547530072.

Subproblem 1
x + 1.412854031 = 1.547530072

Simplifying
x + 1.412854031 = 1.547530072

Reorder the terms:
1.412854031 + x = 1.547530072

Solving
1.412854031 + x = 1.547530072

Solving for variable 'x'.

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

Add '-1.412854031' to each side of the equation.
1.412854031 + -1.412854031 + x = 1.547530072 + -1.412854031

Combine like terms: 1.412854031 + -1.412854031 = 0.000000000
0.000000000 + x = 1.547530072 + -1.412854031
x = 1.547530072 + -1.412854031

Combine like terms: 1.547530072 + -1.412854031 = 0.134676041
x = 0.134676041

Simplifying
x = 0.134676041

Subproblem 2
x + 1.412854031 = -1.547530072

Simplifying
x + 1.412854031 = -1.547530072

Reorder the terms:
1.412854031 + x = -1.547530072

Solving
1.412854031 + x = -1.547530072

Solving for variable 'x'.

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

Add '-1.412854031' to each side of the equation.
1.412854031 + -1.412854031 + x = -1.547530072 + -1.412854031

Combine like terms: 1.412854031 + -1.412854031 = 0.000000000
0.000000000 + x = -1.547530072 + -1.412854031
x = -1.547530072 + -1.412854031

Combine like terms: -1.547530072 + -1.412854031 = -2.960384103
x = -2.960384103

Simplifying
x = -2.960384103

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.134676041, -2.960384103}