6x^2-31x+6=0

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Solution for 6x^2-31x+6=0 equation:


Simplifying
6x2 + -31x + 6 = 0

Reorder the terms:
6 + -31x + 6x2 = 0

Solving
6 + -31x + 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'.
1 + -5.166666667x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5.166666667x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -5.166666667x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5.166666667x + x2 = 0 + -1
-5.166666667x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-5.166666667x + x2 = -1

The x term is -5.166666667x.  Take half its coefficient (-2.583333334).
Square it (6.673611115) and add it to both sides.

Add '6.673611115' to each side of the equation.
-5.166666667x + 6.673611115 + x2 = -1 + 6.673611115

Reorder the terms:
6.673611115 + -5.166666667x + x2 = -1 + 6.673611115

Combine like terms: -1 + 6.673611115 = 5.673611115
6.673611115 + -5.166666667x + x2 = 5.673611115

Factor a perfect square on the left side:
(x + -2.583333334)(x + -2.583333334) = 5.673611115

Calculate the square root of the right side: 2.381934322

Break this problem into two subproblems by setting 
(x + -2.583333334) equal to 2.381934322 and -2.381934322.

Subproblem 1

x + -2.583333334 = 2.381934322 Simplifying x + -2.583333334 = 2.381934322 Reorder the terms: -2.583333334 + x = 2.381934322 Solving -2.583333334 + x = 2.381934322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.583333334' to each side of the equation. -2.583333334 + 2.583333334 + x = 2.381934322 + 2.583333334 Combine like terms: -2.583333334 + 2.583333334 = 0.000000000 0.000000000 + x = 2.381934322 + 2.583333334 x = 2.381934322 + 2.583333334 Combine like terms: 2.381934322 + 2.583333334 = 4.965267656 x = 4.965267656 Simplifying x = 4.965267656

Subproblem 2

x + -2.583333334 = -2.381934322 Simplifying x + -2.583333334 = -2.381934322 Reorder the terms: -2.583333334 + x = -2.381934322 Solving -2.583333334 + x = -2.381934322 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.583333334' to each side of the equation. -2.583333334 + 2.583333334 + x = -2.381934322 + 2.583333334 Combine like terms: -2.583333334 + 2.583333334 = 0.000000000 0.000000000 + x = -2.381934322 + 2.583333334 x = -2.381934322 + 2.583333334 Combine like terms: -2.381934322 + 2.583333334 = 0.201399012 x = 0.201399012 Simplifying x = 0.201399012

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.965267656, 0.201399012}

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