72x^2-180x+78=0

Solution for 72x^2-180x+78=0 equation:

Simplifying
72x2 + -180x + 78 = 0

Reorder the terms:
78 + -180x + 72x2 = 0

Solving
78 + -180x + 72x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(13 + -30x + 12x2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(13 + -30x + 12x2)' equal to zero and attempt to solve:

Simplifying
13 + -30x + 12x2 = 0

Solving
13 + -30x + 12x2 = 0

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

Divide each side by '12'.
1.083333333 + -2.5x + x2 = 0

Move the constant term to the right:

Add '-1.083333333' to each side of the equation.
1.083333333 + -2.5x + -1.083333333 + x2 = 0 + -1.083333333

Reorder the terms:
1.083333333 + -1.083333333 + -2.5x + x2 = 0 + -1.083333333

Combine like terms: 1.083333333 + -1.083333333 = 0.000000000
0.000000000 + -2.5x + x2 = 0 + -1.083333333
-2.5x + x2 = 0 + -1.083333333

Combine like terms: 0 + -1.083333333 = -1.083333333
-2.5x + x2 = -1.083333333

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = -1.083333333 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = -1.083333333 + 1.5625

Combine like terms: -1.083333333 + 1.5625 = 0.479166667
1.5625 + -2.5x + x2 = 0.479166667

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 0.479166667

Calculate the square root of the right side: 0.692218655

Break this problem into two subproblems by setting 
(x + -1.25) equal to 0.692218655 and -0.692218655.

Subproblem 1
x + -1.25 = 0.692218655

Simplifying
x + -1.25 = 0.692218655

Reorder the terms:
-1.25 + x = 0.692218655

Solving
-1.25 + x = 0.692218655

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = 0.692218655 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = 0.692218655 + 1.25
x = 0.692218655 + 1.25

Combine like terms: 0.692218655 + 1.25 = 1.942218655
x = 1.942218655

Simplifying
x = 1.942218655

Subproblem 2
x + -1.25 = -0.692218655

Simplifying
x + -1.25 = -0.692218655

Reorder the terms:
-1.25 + x = -0.692218655

Solving
-1.25 + x = -0.692218655

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = -0.692218655 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = -0.692218655 + 1.25
x = -0.692218655 + 1.25

Combine like terms: -0.692218655 + 1.25 = 0.557781345
x = 0.557781345

Simplifying
x = 0.557781345

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.942218655, 0.557781345}
Solution
x = {1.942218655, 0.557781345}