6x^2+x=82

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Solution for 6x^2+x=82 equation:


Simplifying
6x2 + x = 82

Reorder the terms:
x + 6x2 = 82

Solving
x + 6x2 = 82

Solving for variable 'x'.

Reorder the terms:
-82 + x + 6x2 = 82 + -82

Combine like terms: 82 + -82 = 0
-82 + x + 6x2 = 0

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

Divide each side by '6'.
-13.66666667 + 0.1666666667x + x2 = 0

Move the constant term to the right:

Add '13.66666667' to each side of the equation.
-13.66666667 + 0.1666666667x + 13.66666667 + x2 = 0 + 13.66666667

Reorder the terms:
-13.66666667 + 13.66666667 + 0.1666666667x + x2 = 0 + 13.66666667

Combine like terms: -13.66666667 + 13.66666667 = 0.00000000
0.00000000 + 0.1666666667x + x2 = 0 + 13.66666667
0.1666666667x + x2 = 0 + 13.66666667

Combine like terms: 0 + 13.66666667 = 13.66666667
0.1666666667x + x2 = 13.66666667

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.1666666667x + 0.25 + x2 = 13.66666667 + 0.25

Reorder the terms:
0.25 + 0.1666666667x + x2 = 13.66666667 + 0.25

Combine like terms: 13.66666667 + 0.25 = 13.91666667
0.25 + 0.1666666667x + x2 = 13.91666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 13.91666667

Calculate the square root of the right side: 3.730504881

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.730504881 and -3.730504881.

Subproblem 1

x + 0.5 = 3.730504881 Simplifying x + 0.5 = 3.730504881 Reorder the terms: 0.5 + x = 3.730504881 Solving 0.5 + x = 3.730504881 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 3.730504881 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.730504881 + -0.5 x = 3.730504881 + -0.5 Combine like terms: 3.730504881 + -0.5 = 3.230504881 x = 3.230504881 Simplifying x = 3.230504881

Subproblem 2

x + 0.5 = -3.730504881 Simplifying x + 0.5 = -3.730504881 Reorder the terms: 0.5 + x = -3.730504881 Solving 0.5 + x = -3.730504881 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -3.730504881 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.730504881 + -0.5 x = -3.730504881 + -0.5 Combine like terms: -3.730504881 + -0.5 = -4.230504881 x = -4.230504881 Simplifying x = -4.230504881

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.230504881, -4.230504881}

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