6x^2-6x-31=0

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


Simplifying
6x2 + -6x + -31 = 0

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

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

Move the constant term to the right:

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

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

Combine like terms: -5.166666667 + 5.166666667 = 0.000000000
0.000000000 + -1x + x2 = 0 + 5.166666667
-1x + x2 = 0 + 5.166666667

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

The x term is -1x.  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.
-1x + 0.25 + x2 = 5.166666667 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 5.166666667 + 0.25

Combine like terms: 5.166666667 + 0.25 = 5.416666667
0.25 + -1x + x2 = 5.416666667

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 5.416666667

Calculate the square root of the right side: 2.327373341

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.327373341 and -2.327373341.

Subproblem 1

x + -0.5 = 2.327373341 Simplifying x + -0.5 = 2.327373341 Reorder the terms: -0.5 + x = 2.327373341 Solving -0.5 + x = 2.327373341 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 = 2.327373341 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 2.327373341 + 0.5 x = 2.327373341 + 0.5 Combine like terms: 2.327373341 + 0.5 = 2.827373341 x = 2.827373341 Simplifying x = 2.827373341

Subproblem 2

x + -0.5 = -2.327373341 Simplifying x + -0.5 = -2.327373341 Reorder the terms: -0.5 + x = -2.327373341 Solving -0.5 + x = -2.327373341 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 = -2.327373341 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -2.327373341 + 0.5 x = -2.327373341 + 0.5 Combine like terms: -2.327373341 + 0.5 = -1.827373341 x = -1.827373341 Simplifying x = -1.827373341

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.827373341, -1.827373341}

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