3x=(9x+1)(27-2x)

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Solution for 3x=(9x+1)(27-2x) equation:


Simplifying
3x = (9x + 1)(27 + -2x)

Reorder the terms:
3x = (1 + 9x)(27 + -2x)

Multiply (1 + 9x) * (27 + -2x)
3x = (1(27 + -2x) + 9x * (27 + -2x))
3x = ((27 * 1 + -2x * 1) + 9x * (27 + -2x))
3x = ((27 + -2x) + 9x * (27 + -2x))
3x = (27 + -2x + (27 * 9x + -2x * 9x))
3x = (27 + -2x + (243x + -18x2))

Combine like terms: -2x + 243x = 241x
3x = (27 + 241x + -18x2)

Solving
3x = 27 + 241x + -18x2

Solving for variable 'x'.

Reorder the terms:
-27 + 3x + -241x + 18x2 = 27 + 241x + -18x2 + -27 + -241x + 18x2

Combine like terms: 3x + -241x = -238x
-27 + -238x + 18x2 = 27 + 241x + -18x2 + -27 + -241x + 18x2

Reorder the terms:
-27 + -238x + 18x2 = 27 + -27 + 241x + -241x + -18x2 + 18x2

Combine like terms: 27 + -27 = 0
-27 + -238x + 18x2 = 0 + 241x + -241x + -18x2 + 18x2
-27 + -238x + 18x2 = 241x + -241x + -18x2 + 18x2

Combine like terms: 241x + -241x = 0
-27 + -238x + 18x2 = 0 + -18x2 + 18x2
-27 + -238x + 18x2 = -18x2 + 18x2

Combine like terms: -18x2 + 18x2 = 0
-27 + -238x + 18x2 = 0

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

Divide each side by '18'.
-1.5 + -13.22222222x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -13.22222222x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -13.22222222x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -13.22222222x + x2 = 0 + 1.5
-13.22222222x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-13.22222222x + x2 = 1.5

The x term is -13.22222222x.  Take half its coefficient (-6.61111111).
Square it (43.70679011) and add it to both sides.

Add '43.70679011' to each side of the equation.
-13.22222222x + 43.70679011 + x2 = 1.5 + 43.70679011

Reorder the terms:
43.70679011 + -13.22222222x + x2 = 1.5 + 43.70679011

Combine like terms: 1.5 + 43.70679011 = 45.20679011
43.70679011 + -13.22222222x + x2 = 45.20679011

Factor a perfect square on the left side:
(x + -6.61111111)(x + -6.61111111) = 45.20679011

Calculate the square root of the right side: 6.723599491

Break this problem into two subproblems by setting 
(x + -6.61111111) equal to 6.723599491 and -6.723599491.

Subproblem 1

x + -6.61111111 = 6.723599491 Simplifying x + -6.61111111 = 6.723599491 Reorder the terms: -6.61111111 + x = 6.723599491 Solving -6.61111111 + x = 6.723599491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.61111111' to each side of the equation. -6.61111111 + 6.61111111 + x = 6.723599491 + 6.61111111 Combine like terms: -6.61111111 + 6.61111111 = 0.00000000 0.00000000 + x = 6.723599491 + 6.61111111 x = 6.723599491 + 6.61111111 Combine like terms: 6.723599491 + 6.61111111 = 13.334710601 x = 13.334710601 Simplifying x = 13.334710601

Subproblem 2

x + -6.61111111 = -6.723599491 Simplifying x + -6.61111111 = -6.723599491 Reorder the terms: -6.61111111 + x = -6.723599491 Solving -6.61111111 + x = -6.723599491 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '6.61111111' to each side of the equation. -6.61111111 + 6.61111111 + x = -6.723599491 + 6.61111111 Combine like terms: -6.61111111 + 6.61111111 = 0.00000000 0.00000000 + x = -6.723599491 + 6.61111111 x = -6.723599491 + 6.61111111 Combine like terms: -6.723599491 + 6.61111111 = -0.112488381 x = -0.112488381 Simplifying x = -0.112488381

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13.334710601, -0.112488381}

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