(180-x)-6x(90-x)=50

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Solution for (180-x)-6x(90-x)=50 equation:


Simplifying
(180 + -1x) + -6x(90 + -1x) = 50

Remove parenthesis around (180 + -1x)
180 + -1x + -6x(90 + -1x) = 50
180 + -1x + (90 * -6x + -1x * -6x) = 50
180 + -1x + (-540x + 6x2) = 50

Combine like terms: -1x + -540x = -541x
180 + -541x + 6x2 = 50

Solving
180 + -541x + 6x2 = 50

Solving for variable 'x'.

Reorder the terms:
180 + -50 + -541x + 6x2 = 50 + -50

Combine like terms: 180 + -50 = 130
130 + -541x + 6x2 = 50 + -50

Combine like terms: 50 + -50 = 0
130 + -541x + 6x2 = 0

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

Divide each side by '6'.
21.66666667 + -90.16666667x + x2 = 0

Move the constant term to the right:

Add '-21.66666667' to each side of the equation.
21.66666667 + -90.16666667x + -21.66666667 + x2 = 0 + -21.66666667

Reorder the terms:
21.66666667 + -21.66666667 + -90.16666667x + x2 = 0 + -21.66666667

Combine like terms: 21.66666667 + -21.66666667 = 0.00000000
0.00000000 + -90.16666667x + x2 = 0 + -21.66666667
-90.16666667x + x2 = 0 + -21.66666667

Combine like terms: 0 + -21.66666667 = -21.66666667
-90.16666667x + x2 = -21.66666667

The x term is -90.16666667x.  Take half its coefficient (-45.08333334).
Square it (2032.506945) and add it to both sides.

Add '2032.506945' to each side of the equation.
-90.16666667x + 2032.506945 + x2 = -21.66666667 + 2032.506945

Reorder the terms:
2032.506945 + -90.16666667x + x2 = -21.66666667 + 2032.506945

Combine like terms: -21.66666667 + 2032.506945 = 2010.84027833
2032.506945 + -90.16666667x + x2 = 2010.84027833

Factor a perfect square on the left side:
(x + -45.08333334)(x + -45.08333334) = 2010.84027833

Calculate the square root of the right side: 44.842393762

Break this problem into two subproblems by setting 
(x + -45.08333334) equal to 44.842393762 and -44.842393762.

Subproblem 1

x + -45.08333334 = 44.842393762 Simplifying x + -45.08333334 = 44.842393762 Reorder the terms: -45.08333334 + x = 44.842393762 Solving -45.08333334 + x = 44.842393762 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45.08333334' to each side of the equation. -45.08333334 + 45.08333334 + x = 44.842393762 + 45.08333334 Combine like terms: -45.08333334 + 45.08333334 = 0.00000000 0.00000000 + x = 44.842393762 + 45.08333334 x = 44.842393762 + 45.08333334 Combine like terms: 44.842393762 + 45.08333334 = 89.925727102 x = 89.925727102 Simplifying x = 89.925727102

Subproblem 2

x + -45.08333334 = -44.842393762 Simplifying x + -45.08333334 = -44.842393762 Reorder the terms: -45.08333334 + x = -44.842393762 Solving -45.08333334 + x = -44.842393762 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45.08333334' to each side of the equation. -45.08333334 + 45.08333334 + x = -44.842393762 + 45.08333334 Combine like terms: -45.08333334 + 45.08333334 = 0.00000000 0.00000000 + x = -44.842393762 + 45.08333334 x = -44.842393762 + 45.08333334 Combine like terms: -44.842393762 + 45.08333334 = 0.240939578 x = 0.240939578 Simplifying x = 0.240939578

Solution

The solution to the problem is based on the solutions from the subproblems. x = {89.925727102, 0.240939578}

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