120x=x^2-60x+90

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Solution for 120x=x^2-60x+90 equation:


Simplifying
120x = x2 + -60x + 90

Reorder the terms:
120x = 90 + -60x + x2

Solving
120x = 90 + -60x + x2

Solving for variable 'x'.

Reorder the terms:
-90 + 120x + 60x + -1x2 = 90 + -60x + x2 + -90 + 60x + -1x2

Combine like terms: 120x + 60x = 180x
-90 + 180x + -1x2 = 90 + -60x + x2 + -90 + 60x + -1x2

Reorder the terms:
-90 + 180x + -1x2 = 90 + -90 + -60x + 60x + x2 + -1x2

Combine like terms: 90 + -90 = 0
-90 + 180x + -1x2 = 0 + -60x + 60x + x2 + -1x2
-90 + 180x + -1x2 = -60x + 60x + x2 + -1x2

Combine like terms: -60x + 60x = 0
-90 + 180x + -1x2 = 0 + x2 + -1x2
-90 + 180x + -1x2 = x2 + -1x2

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

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
90 + -180x + x2 = 0

Move the constant term to the right:

Add '-90' to each side of the equation.
90 + -180x + -90 + x2 = 0 + -90

Reorder the terms:
90 + -90 + -180x + x2 = 0 + -90

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

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

The x term is -180x.  Take half its coefficient (-90).
Square it (8100) and add it to both sides.

Add '8100' to each side of the equation.
-180x + 8100 + x2 = -90 + 8100

Reorder the terms:
8100 + -180x + x2 = -90 + 8100

Combine like terms: -90 + 8100 = 8010
8100 + -180x + x2 = 8010

Factor a perfect square on the left side:
(x + -90)(x + -90) = 8010

Calculate the square root of the right side: 89.498603341

Break this problem into two subproblems by setting 
(x + -90) equal to 89.498603341 and -89.498603341.

Subproblem 1

x + -90 = 89.498603341 Simplifying x + -90 = 89.498603341 Reorder the terms: -90 + x = 89.498603341 Solving -90 + x = 89.498603341 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '90' to each side of the equation. -90 + 90 + x = 89.498603341 + 90 Combine like terms: -90 + 90 = 0 0 + x = 89.498603341 + 90 x = 89.498603341 + 90 Combine like terms: 89.498603341 + 90 = 179.498603341 x = 179.498603341 Simplifying x = 179.498603341

Subproblem 2

x + -90 = -89.498603341 Simplifying x + -90 = -89.498603341 Reorder the terms: -90 + x = -89.498603341 Solving -90 + x = -89.498603341 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '90' to each side of the equation. -90 + 90 + x = -89.498603341 + 90 Combine like terms: -90 + 90 = 0 0 + x = -89.498603341 + 90 x = -89.498603341 + 90 Combine like terms: -89.498603341 + 90 = 0.501396659 x = 0.501396659 Simplifying x = 0.501396659

Solution

The solution to the problem is based on the solutions from the subproblems. x = {179.498603341, 0.501396659}

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