6x^2+12x-31=0

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Solution for 6x^2+12x-31=0 equation:


Simplifying
6x2 + 12x + -31 = 0

Reorder the terms:
-31 + 12x + 6x2 = 0

Solving
-31 + 12x + 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 + 2x + x2 = 0

Move the constant term to the right:

Add '5.166666667' to each side of the equation.
-5.166666667 + 2x + 5.166666667 + x2 = 0 + 5.166666667

Reorder the terms:
-5.166666667 + 5.166666667 + 2x + x2 = 0 + 5.166666667

Combine like terms: -5.166666667 + 5.166666667 = 0.000000000
0.000000000 + 2x + x2 = 0 + 5.166666667
2x + x2 = 0 + 5.166666667

Combine like terms: 0 + 5.166666667 = 5.166666667
2x + x2 = 5.166666667

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 5.166666667 + 1

Reorder the terms:
1 + 2x + x2 = 5.166666667 + 1

Combine like terms: 5.166666667 + 1 = 6.166666667
1 + 2x + x2 = 6.166666667

Factor a perfect square on the left side:
(x + 1)(x + 1) = 6.166666667

Calculate the square root of the right side: 2.483277404

Break this problem into two subproblems by setting 
(x + 1) equal to 2.483277404 and -2.483277404.

Subproblem 1

x + 1 = 2.483277404 Simplifying x + 1 = 2.483277404 Reorder the terms: 1 + x = 2.483277404 Solving 1 + x = 2.483277404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.483277404 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.483277404 + -1 x = 2.483277404 + -1 Combine like terms: 2.483277404 + -1 = 1.483277404 x = 1.483277404 Simplifying x = 1.483277404

Subproblem 2

x + 1 = -2.483277404 Simplifying x + 1 = -2.483277404 Reorder the terms: 1 + x = -2.483277404 Solving 1 + x = -2.483277404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.483277404 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.483277404 + -1 x = -2.483277404 + -1 Combine like terms: -2.483277404 + -1 = -3.483277404 x = -3.483277404 Simplifying x = -3.483277404

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.483277404, -3.483277404}

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