-54x^2+180x-125=0

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Solution for -54x^2+180x-125=0 equation: 


Simplifying
-54x2 + 180x + -125 = 0

Reorder the terms:
-125 + 180x + -54x2 = 0

Solving
-125 + 180x + -54x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-54 the coefficient of the squared term: 

Divide each side by '-54'.
2.314814815 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '-2.314814815' to each side of the equation.
2.314814815 + -3.333333333x + -2.314814815 + x2 = 0 + -2.314814815

Reorder the terms:
2.314814815 + -2.314814815 + -3.333333333x + x2 = 0 + -2.314814815

Combine like terms: 2.314814815 + -2.314814815 = 0.000000000
0.000000000 + -3.333333333x + x2 = 0 + -2.314814815
-3.333333333x + x2 = 0 + -2.314814815

Combine like terms: 0 + -2.314814815 = -2.314814815
-3.333333333x + x2 = -2.314814815

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = -2.314814815 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = -2.314814815 + 2.777777779

Combine like terms: -2.314814815 + 2.777777779 = 0.462962964
2.777777779 + -3.333333333x + x2 = 0.462962964

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 0.462962964

Calculate the square root of the right side: 0.680413818

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 0.680413818 and -0.680413818.


Subproblem 1
x + -1.666666667 = 0.680413818

Simplifying
x + -1.666666667 = 0.680413818

Reorder the terms:
-1.666666667 + x = 0.680413818

Solving
-1.666666667 + x = 0.680413818

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = 0.680413818 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = 0.680413818 + 1.666666667
x = 0.680413818 + 1.666666667

Combine like terms: 0.680413818 + 1.666666667 = 2.347080485
x = 2.347080485

Simplifying
x = 2.347080485


Subproblem 2
x + -1.666666667 = -0.680413818

Simplifying
x + -1.666666667 = -0.680413818

Reorder the terms:
-1.666666667 + x = -0.680413818

Solving
-1.666666667 + x = -0.680413818

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = -0.680413818 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = -0.680413818 + 1.666666667
x = -0.680413818 + 1.666666667

Combine like terms: -0.680413818 + 1.666666667 = 0.986252849
x = 0.986252849

Simplifying
x = 0.986252849


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.347080485, 0.986252849}

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