12x^2-6x-23=0

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Solution for 12x^2-6x-23=0 equation:


Simplifying
12x2 + -6x + -23 = 0

Reorder the terms:
-23 + -6x + 12x2 = 0

Solving
-23 + -6x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
-1.916666667 + -0.5x + x2 = 0

Move the constant term to the right:

Add '1.916666667' to each side of the equation.
-1.916666667 + -0.5x + 1.916666667 + x2 = 0 + 1.916666667

Reorder the terms:
-1.916666667 + 1.916666667 + -0.5x + x2 = 0 + 1.916666667

Combine like terms: -1.916666667 + 1.916666667 = 0.000000000
0.000000000 + -0.5x + x2 = 0 + 1.916666667
-0.5x + x2 = 0 + 1.916666667

Combine like terms: 0 + 1.916666667 = 1.916666667
-0.5x + x2 = 1.916666667

The x term is -0.5x.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5x + 0.0625 + x2 = 1.916666667 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 1.916666667 + 0.0625

Combine like terms: 1.916666667 + 0.0625 = 1.979166667
0.0625 + -0.5x + x2 = 1.979166667

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 1.979166667

Calculate the square root of the right side: 1.406828585

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.406828585 and -1.406828585.

Subproblem 1

x + -0.25 = 1.406828585 Simplifying x + -0.25 = 1.406828585 Reorder the terms: -0.25 + x = 1.406828585 Solving -0.25 + x = 1.406828585 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 1.406828585 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 1.406828585 + 0.25 x = 1.406828585 + 0.25 Combine like terms: 1.406828585 + 0.25 = 1.656828585 x = 1.656828585 Simplifying x = 1.656828585

Subproblem 2

x + -0.25 = -1.406828585 Simplifying x + -0.25 = -1.406828585 Reorder the terms: -0.25 + x = -1.406828585 Solving -0.25 + x = -1.406828585 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -1.406828585 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -1.406828585 + 0.25 x = -1.406828585 + 0.25 Combine like terms: -1.406828585 + 0.25 = -1.156828585 x = -1.156828585 Simplifying x = -1.156828585

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.656828585, -1.156828585}

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