6x^2-30x+13=0

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Solution for 6x^2-30x+13=0 equation: 


Simplifying
6x2 + -30x + 13 = 0

Reorder the terms:
13 + -30x + 6x2 = 0

Solving
13 + -30x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
2.166666667 + -5x + x2 = 0

Move the constant term to the right:

Add '-2.166666667' to each side of the equation.
2.166666667 + -5x + -2.166666667 + x2 = 0 + -2.166666667

Reorder the terms:
2.166666667 + -2.166666667 + -5x + x2 = 0 + -2.166666667

Combine like terms: 2.166666667 + -2.166666667 = 0.000000000
0.000000000 + -5x + x2 = 0 + -2.166666667
-5x + x2 = 0 + -2.166666667

Combine like terms: 0 + -2.166666667 = -2.166666667
-5x + x2 = -2.166666667

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -2.166666667 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -2.166666667 + 6.25

Combine like terms: -2.166666667 + 6.25 = 4.083333333
6.25 + -5x + x2 = 4.083333333

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 4.083333333

Calculate the square root of the right side: 2.020725942

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.020725942 and -2.020725942.


Subproblem 1
x + -2.5 = 2.020725942

Simplifying
x + -2.5 = 2.020725942

Reorder the terms:
-2.5 + x = 2.020725942

Solving
-2.5 + x = 2.020725942

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 2.020725942 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 2.020725942 + 2.5
x = 2.020725942 + 2.5

Combine like terms: 2.020725942 + 2.5 = 4.520725942
x = 4.520725942

Simplifying
x = 4.520725942


Subproblem 2
x + -2.5 = -2.020725942

Simplifying
x + -2.5 = -2.020725942

Reorder the terms:
-2.5 + x = -2.020725942

Solving
-2.5 + x = -2.020725942

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -2.020725942 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -2.020725942 + 2.5
x = -2.020725942 + 2.5

Combine like terms: -2.020725942 + 2.5 = 0.479274058
x = 0.479274058

Simplifying
x = 0.479274058


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.520725942, 0.479274058}

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