180-x=x(90-x)

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


Simplifying
180 + -1x = x(90 + -1x)
180 + -1x = (90 * x + -1x * x)
180 + -1x = (90x + -1x2)

Solving
180 + -1x = 90x + -1x2

Solving for variable 'x'.

Combine like terms: -1x + -90x = -91x
180 + -91x + x2 = 90x + -1x2 + -90x + x2

Reorder the terms:
180 + -91x + x2 = 90x + -90x + -1x2 + x2

Combine like terms: 90x + -90x = 0
180 + -91x + x2 = 0 + -1x2 + x2
180 + -91x + x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
180 + -91x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-180' to each side of the equation.
180 + -91x + -180 + x2 = 0 + -180

Reorder the terms:
180 + -180 + -91x + x2 = 0 + -180

Combine like terms: 180 + -180 = 0
0 + -91x + x2 = 0 + -180
-91x + x2 = 0 + -180

Combine like terms: 0 + -180 = -180
-91x + x2 = -180

The x term is -91x.  Take half its coefficient (-45.5).
Square it (2070.25) and add it to both sides.

Add '2070.25' to each side of the equation.
-91x + 2070.25 + x2 = -180 + 2070.25

Reorder the terms:
2070.25 + -91x + x2 = -180 + 2070.25

Combine like terms: -180 + 2070.25 = 1890.25
2070.25 + -91x + x2 = 1890.25

Factor a perfect square on the left side:
(x + -45.5)(x + -45.5) = 1890.25

Calculate the square root of the right side: 43.477005417

Break this problem into two subproblems by setting 
(x + -45.5) equal to 43.477005417 and -43.477005417.

Subproblem 1

x + -45.5 = 43.477005417 Simplifying x + -45.5 = 43.477005417 Reorder the terms: -45.5 + x = 43.477005417 Solving -45.5 + x = 43.477005417 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45.5' to each side of the equation. -45.5 + 45.5 + x = 43.477005417 + 45.5 Combine like terms: -45.5 + 45.5 = 0.0 0.0 + x = 43.477005417 + 45.5 x = 43.477005417 + 45.5 Combine like terms: 43.477005417 + 45.5 = 88.977005417 x = 88.977005417 Simplifying x = 88.977005417

Subproblem 2

x + -45.5 = -43.477005417 Simplifying x + -45.5 = -43.477005417 Reorder the terms: -45.5 + x = -43.477005417 Solving -45.5 + x = -43.477005417 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '45.5' to each side of the equation. -45.5 + 45.5 + x = -43.477005417 + 45.5 Combine like terms: -45.5 + 45.5 = 0.0 0.0 + x = -43.477005417 + 45.5 x = -43.477005417 + 45.5 Combine like terms: -43.477005417 + 45.5 = 2.022994583 x = 2.022994583 Simplifying x = 2.022994583

Solution

The solution to the problem is based on the solutions from the subproblems. x = {88.977005417, 2.022994583}

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