6x^2-13x-9=0

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Solution for 6x^2-13x-9=0 equation:


Simplifying
6x2 + -13x + -9 = 0

Reorder the terms:
-9 + -13x + 6x2 = 0

Solving
-9 + -13x + 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'.
-1.5 + -2.166666667x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -2.166666667x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -2.166666667x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -2.166666667x + x2 = 0 + 1.5
-2.166666667x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-2.166666667x + x2 = 1.5

The x term is -2.166666667x.  Take half its coefficient (-1.083333334).
Square it (1.173611113) and add it to both sides.

Add '1.173611113' to each side of the equation.
-2.166666667x + 1.173611113 + x2 = 1.5 + 1.173611113

Reorder the terms:
1.173611113 + -2.166666667x + x2 = 1.5 + 1.173611113

Combine like terms: 1.5 + 1.173611113 = 2.673611113
1.173611113 + -2.166666667x + x2 = 2.673611113

Factor a perfect square on the left side:
(x + -1.083333334)(x + -1.083333334) = 2.673611113

Calculate the square root of the right side: 1.635118073

Break this problem into two subproblems by setting 
(x + -1.083333334) equal to 1.635118073 and -1.635118073.

Subproblem 1

x + -1.083333334 = 1.635118073 Simplifying x + -1.083333334 = 1.635118073 Reorder the terms: -1.083333334 + x = 1.635118073 Solving -1.083333334 + x = 1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + x = 1.635118073 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = 1.635118073 + 1.083333334 x = 1.635118073 + 1.083333334 Combine like terms: 1.635118073 + 1.083333334 = 2.718451407 x = 2.718451407 Simplifying x = 2.718451407

Subproblem 2

x + -1.083333334 = -1.635118073 Simplifying x + -1.083333334 = -1.635118073 Reorder the terms: -1.083333334 + x = -1.635118073 Solving -1.083333334 + x = -1.635118073 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + x = -1.635118073 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = -1.635118073 + 1.083333334 x = -1.635118073 + 1.083333334 Combine like terms: -1.635118073 + 1.083333334 = -0.551784739 x = -0.551784739 Simplifying x = -0.551784739

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.718451407, -0.551784739}

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