6x^2-41x+16=0

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Solution for 6x^2-41x+16=0 equation: 


Simplifying
6x2 + -41x + 16 = 0

Reorder the terms:
16 + -41x + 6x2 = 0

Solving
16 + -41x + 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'.
2.666666667 + -6.833333333x + x2 = 0

Move the constant term to the right:

Add '-2.666666667' to each side of the equation.
2.666666667 + -6.833333333x + -2.666666667 + x2 = 0 + -2.666666667

Reorder the terms:
2.666666667 + -2.666666667 + -6.833333333x + x2 = 0 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + -6.833333333x + x2 = 0 + -2.666666667
-6.833333333x + x2 = 0 + -2.666666667

Combine like terms: 0 + -2.666666667 = -2.666666667
-6.833333333x + x2 = -2.666666667

The x term is -6.833333333x.  Take half its coefficient (-3.416666667).
Square it (11.67361111) and add it to both sides.

Add '11.67361111' to each side of the equation.
-6.833333333x + 11.67361111 + x2 = -2.666666667 + 11.67361111

Reorder the terms:
11.67361111 + -6.833333333x + x2 = -2.666666667 + 11.67361111

Combine like terms: -2.666666667 + 11.67361111 = 9.006944443
11.67361111 + -6.833333333x + x2 = 9.006944443

Factor a perfect square on the left side:
(x + -3.416666667)(x + -3.416666667) = 9.006944443

Calculate the square root of the right side: 3.001157184

Break this problem into two subproblems by setting 
(x + -3.416666667) equal to 3.001157184 and -3.001157184.


Subproblem 1
x + -3.416666667 = 3.001157184

Simplifying
x + -3.416666667 = 3.001157184

Reorder the terms:
-3.416666667 + x = 3.001157184

Solving
-3.416666667 + x = 3.001157184

Solving for variable 'x'.

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

Add '3.416666667' to each side of the equation.
-3.416666667 + 3.416666667 + x = 3.001157184 + 3.416666667

Combine like terms: -3.416666667 + 3.416666667 = 0.000000000
0.000000000 + x = 3.001157184 + 3.416666667
x = 3.001157184 + 3.416666667

Combine like terms: 3.001157184 + 3.416666667 = 6.417823851
x = 6.417823851

Simplifying
x = 6.417823851


Subproblem 2
x + -3.416666667 = -3.001157184

Simplifying
x + -3.416666667 = -3.001157184

Reorder the terms:
-3.416666667 + x = -3.001157184

Solving
-3.416666667 + x = -3.001157184

Solving for variable 'x'.

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

Add '3.416666667' to each side of the equation.
-3.416666667 + 3.416666667 + x = -3.001157184 + 3.416666667

Combine like terms: -3.416666667 + 3.416666667 = 0.000000000
0.000000000 + x = -3.001157184 + 3.416666667
x = -3.001157184 + 3.416666667

Combine like terms: -3.001157184 + 3.416666667 = 0.415509483
x = 0.415509483

Simplifying
x = 0.415509483


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.417823851, 0.415509483}

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