12x^2-13x-1=0

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Solution for 12x^2-13x-1=0 equation:


Simplifying
12x2 + -13x + -1 = 0

Reorder the terms:
-1 + -13x + 12x2 = 0

Solving
-1 + -13x + 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'.
-0.08333333333 + -1.083333333x + x2 = 0

Move the constant term to the right:

Add '0.08333333333' to each side of the equation.
-0.08333333333 + -1.083333333x + 0.08333333333 + x2 = 0 + 0.08333333333

Reorder the terms:
-0.08333333333 + 0.08333333333 + -1.083333333x + x2 = 0 + 0.08333333333

Combine like terms: -0.08333333333 + 0.08333333333 = 0.00000000000
0.00000000000 + -1.083333333x + x2 = 0 + 0.08333333333
-1.083333333x + x2 = 0 + 0.08333333333

Combine like terms: 0 + 0.08333333333 = 0.08333333333
-1.083333333x + x2 = 0.08333333333

The x term is -1.083333333x.  Take half its coefficient (-0.5416666665).
Square it (0.2934027776) and add it to both sides.

Add '0.2934027776' to each side of the equation.
-1.083333333x + 0.2934027776 + x2 = 0.08333333333 + 0.2934027776

Reorder the terms:
0.2934027776 + -1.083333333x + x2 = 0.08333333333 + 0.2934027776

Combine like terms: 0.08333333333 + 0.2934027776 = 0.37673611093
0.2934027776 + -1.083333333x + x2 = 0.37673611093

Factor a perfect square on the left side:
(x + -0.5416666665)(x + -0.5416666665) = 0.37673611093

Calculate the square root of the right side: 0.613788327

Break this problem into two subproblems by setting 
(x + -0.5416666665) equal to 0.613788327 and -0.613788327.

Subproblem 1

x + -0.5416666665 = 0.613788327 Simplifying x + -0.5416666665 = 0.613788327 Reorder the terms: -0.5416666665 + x = 0.613788327 Solving -0.5416666665 + x = 0.613788327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + x = 0.613788327 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = 0.613788327 + 0.5416666665 x = 0.613788327 + 0.5416666665 Combine like terms: 0.613788327 + 0.5416666665 = 1.1554549935 x = 1.1554549935 Simplifying x = 1.1554549935

Subproblem 2

x + -0.5416666665 = -0.613788327 Simplifying x + -0.5416666665 = -0.613788327 Reorder the terms: -0.5416666665 + x = -0.613788327 Solving -0.5416666665 + x = -0.613788327 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + x = -0.613788327 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = -0.613788327 + 0.5416666665 x = -0.613788327 + 0.5416666665 Combine like terms: -0.613788327 + 0.5416666665 = -0.0721216605 x = -0.0721216605 Simplifying x = -0.0721216605

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.1554549935, -0.0721216605}

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