-6x^2+39x-61=0

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Solution for -6x^2+39x-61=0 equation: 


Simplifying
-6x2 + 39x + -61 = 0

Reorder the terms:
-61 + 39x + -6x2 = 0

Solving
-61 + 39x + -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'.
10.16666667 + -6.5x + x2 = 0

Move the constant term to the right:

Add '-10.16666667' to each side of the equation.
10.16666667 + -6.5x + -10.16666667 + x2 = 0 + -10.16666667

Reorder the terms:
10.16666667 + -10.16666667 + -6.5x + x2 = 0 + -10.16666667

Combine like terms: 10.16666667 + -10.16666667 = 0.00000000
0.00000000 + -6.5x + x2 = 0 + -10.16666667
-6.5x + x2 = 0 + -10.16666667

Combine like terms: 0 + -10.16666667 = -10.16666667
-6.5x + x2 = -10.16666667

The x term is -6.5x.  Take half its coefficient (-3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
-6.5x + 10.5625 + x2 = -10.16666667 + 10.5625

Reorder the terms:
10.5625 + -6.5x + x2 = -10.16666667 + 10.5625

Combine like terms: -10.16666667 + 10.5625 = 0.39583333
10.5625 + -6.5x + x2 = 0.39583333

Factor a perfect square on the left side:
(x + -3.25)(x + -3.25) = 0.39583333

Calculate the square root of the right side: 0.629152867

Break this problem into two subproblems by setting 
(x + -3.25) equal to 0.629152867 and -0.629152867.


Subproblem 1
x + -3.25 = 0.629152867

Simplifying
x + -3.25 = 0.629152867

Reorder the terms:
-3.25 + x = 0.629152867

Solving
-3.25 + x = 0.629152867

Solving for variable 'x'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + x = 0.629152867 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + x = 0.629152867 + 3.25
x = 0.629152867 + 3.25

Combine like terms: 0.629152867 + 3.25 = 3.879152867
x = 3.879152867

Simplifying
x = 3.879152867


Subproblem 2
x + -3.25 = -0.629152867

Simplifying
x + -3.25 = -0.629152867

Reorder the terms:
-3.25 + x = -0.629152867

Solving
-3.25 + x = -0.629152867

Solving for variable 'x'.

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

Add '3.25' to each side of the equation.
-3.25 + 3.25 + x = -0.629152867 + 3.25

Combine like terms: -3.25 + 3.25 = 0.00
0.00 + x = -0.629152867 + 3.25
x = -0.629152867 + 3.25

Combine like terms: -0.629152867 + 3.25 = 2.620847133
x = 2.620847133

Simplifying
x = 2.620847133


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.879152867, 2.620847133}

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