67-x=3x(43-x)

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Solution for 67-x=3x(43-x) equation:


Simplifying
67 + -1x = 3x(43 + -1x)
67 + -1x = (43 * 3x + -1x * 3x)
67 + -1x = (129x + -3x2)

Solving
67 + -1x = 129x + -3x2

Solving for variable 'x'.

Combine like terms: -1x + -129x = -130x
67 + -130x + 3x2 = 129x + -3x2 + -129x + 3x2

Reorder the terms:
67 + -130x + 3x2 = 129x + -129x + -3x2 + 3x2

Combine like terms: 129x + -129x = 0
67 + -130x + 3x2 = 0 + -3x2 + 3x2
67 + -130x + 3x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
67 + -130x + 3x2 = 0

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

Divide each side by '3'.
22.33333333 + -43.33333333x + x2 = 0

Move the constant term to the right:

Add '-22.33333333' to each side of the equation.
22.33333333 + -43.33333333x + -22.33333333 + x2 = 0 + -22.33333333

Reorder the terms:
22.33333333 + -22.33333333 + -43.33333333x + x2 = 0 + -22.33333333

Combine like terms: 22.33333333 + -22.33333333 = 0.00000000
0.00000000 + -43.33333333x + x2 = 0 + -22.33333333
-43.33333333x + x2 = 0 + -22.33333333

Combine like terms: 0 + -22.33333333 = -22.33333333
-43.33333333x + x2 = -22.33333333

The x term is -43.33333333x.  Take half its coefficient (-21.66666667).
Square it (469.4444446) and add it to both sides.

Add '469.4444446' to each side of the equation.
-43.33333333x + 469.4444446 + x2 = -22.33333333 + 469.4444446

Reorder the terms:
469.4444446 + -43.33333333x + x2 = -22.33333333 + 469.4444446

Combine like terms: -22.33333333 + 469.4444446 = 447.11111127
469.4444446 + -43.33333333x + x2 = 447.11111127

Factor a perfect square on the left side:
(x + -21.66666667)(x + -21.66666667) = 447.11111127

Calculate the square root of the right side: 21.14500204

Break this problem into two subproblems by setting 
(x + -21.66666667) equal to 21.14500204 and -21.14500204.

Subproblem 1

x + -21.66666667 = 21.14500204 Simplifying x + -21.66666667 = 21.14500204 Reorder the terms: -21.66666667 + x = 21.14500204 Solving -21.66666667 + x = 21.14500204 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21.66666667' to each side of the equation. -21.66666667 + 21.66666667 + x = 21.14500204 + 21.66666667 Combine like terms: -21.66666667 + 21.66666667 = 0.00000000 0.00000000 + x = 21.14500204 + 21.66666667 x = 21.14500204 + 21.66666667 Combine like terms: 21.14500204 + 21.66666667 = 42.81166871 x = 42.81166871 Simplifying x = 42.81166871

Subproblem 2

x + -21.66666667 = -21.14500204 Simplifying x + -21.66666667 = -21.14500204 Reorder the terms: -21.66666667 + x = -21.14500204 Solving -21.66666667 + x = -21.14500204 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21.66666667' to each side of the equation. -21.66666667 + 21.66666667 + x = -21.14500204 + 21.66666667 Combine like terms: -21.66666667 + 21.66666667 = 0.00000000 0.00000000 + x = -21.14500204 + 21.66666667 x = -21.14500204 + 21.66666667 Combine like terms: -21.14500204 + 21.66666667 = 0.52166463 x = 0.52166463 Simplifying x = 0.52166463

Solution

The solution to the problem is based on the solutions from the subproblems. x = {42.81166871, 0.52166463}

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