12x^2-13x-4=1

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Solution for 12x^2-13x-4=1 equation:


Simplifying
12x2 + -13x + -4 = 1

Reorder the terms:
-4 + -13x + 12x2 = 1

Solving
-4 + -13x + 12x2 = 1

Solving for variable 'x'.

Reorder the terms:
-4 + -1 + -13x + 12x2 = 1 + -1

Combine like terms: -4 + -1 = -5
-5 + -13x + 12x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-5 + -13x + 12x2 = 0

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

Divide each side by '12'.
-0.4166666667 + -1.083333333x + x2 = 0

Move the constant term to the right:

Add '0.4166666667' to each side of the equation.
-0.4166666667 + -1.083333333x + 0.4166666667 + x2 = 0 + 0.4166666667

Reorder the terms:
-0.4166666667 + 0.4166666667 + -1.083333333x + x2 = 0 + 0.4166666667

Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000
0.0000000000 + -1.083333333x + x2 = 0 + 0.4166666667
-1.083333333x + x2 = 0 + 0.4166666667

Combine like terms: 0 + 0.4166666667 = 0.4166666667
-1.083333333x + x2 = 0.4166666667

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 = 0.4166666667 + 0.2934027776

Reorder the terms:
0.2934027776 + -1.083333333x + x2 = 0.4166666667 + 0.2934027776

Combine like terms: 0.4166666667 + 0.2934027776 = 0.7100694443
0.2934027776 + -1.083333333x + x2 = 0.7100694443

Factor a perfect square on the left side:
(x + -0.5416666665)(x + -0.5416666665) = 0.7100694443

Calculate the square root of the right side: 0.842656184

Break this problem into two subproblems by setting 
(x + -0.5416666665) equal to 0.842656184 and -0.842656184.

Subproblem 1

x + -0.5416666665 = 0.842656184 Simplifying x + -0.5416666665 = 0.842656184 Reorder the terms: -0.5416666665 + x = 0.842656184 Solving -0.5416666665 + x = 0.842656184 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 = 0.842656184 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = 0.842656184 + 0.5416666665 x = 0.842656184 + 0.5416666665 Combine like terms: 0.842656184 + 0.5416666665 = 1.3843228505 x = 1.3843228505 Simplifying x = 1.3843228505

Subproblem 2

x + -0.5416666665 = -0.842656184 Simplifying x + -0.5416666665 = -0.842656184 Reorder the terms: -0.5416666665 + x = -0.842656184 Solving -0.5416666665 + x = -0.842656184 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 = -0.842656184 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + x = -0.842656184 + 0.5416666665 x = -0.842656184 + 0.5416666665 Combine like terms: -0.842656184 + 0.5416666665 = -0.3009895175 x = -0.3009895175 Simplifying x = -0.3009895175

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.3843228505, -0.3009895175}

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