6x^2-31x-32=0

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Solution for 6x^2-31x-32=0 equation: 


Simplifying
6x2 + -31x + -32 = 0

Reorder the terms:
-32 + -31x + 6x2 = 0

Solving
-32 + -31x + 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'.
-5.333333333 + -5.166666667x + x2 = 0

Move the constant term to the right:

Add '5.333333333' to each side of the equation.
-5.333333333 + -5.166666667x + 5.333333333 + x2 = 0 + 5.333333333

Reorder the terms:
-5.333333333 + 5.333333333 + -5.166666667x + x2 = 0 + 5.333333333

Combine like terms: -5.333333333 + 5.333333333 = 0.000000000
0.000000000 + -5.166666667x + x2 = 0 + 5.333333333
-5.166666667x + x2 = 0 + 5.333333333

Combine like terms: 0 + 5.333333333 = 5.333333333
-5.166666667x + x2 = 5.333333333

The x term is -5.166666667x.  Take half its coefficient (-2.583333334).
Square it (6.673611115) and add it to both sides.

Add '6.673611115' to each side of the equation.
-5.166666667x + 6.673611115 + x2 = 5.333333333 + 6.673611115

Reorder the terms:
6.673611115 + -5.166666667x + x2 = 5.333333333 + 6.673611115

Combine like terms: 5.333333333 + 6.673611115 = 12.006944448
6.673611115 + -5.166666667x + x2 = 12.006944448

Factor a perfect square on the left side:
(x + -2.583333334)(x + -2.583333334) = 12.006944448

Calculate the square root of the right side: 3.465103815

Break this problem into two subproblems by setting 
(x + -2.583333334) equal to 3.465103815 and -3.465103815.


Subproblem 1
x + -2.583333334 = 3.465103815

Simplifying
x + -2.583333334 = 3.465103815

Reorder the terms:
-2.583333334 + x = 3.465103815

Solving
-2.583333334 + x = 3.465103815

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.583333334' to each side of the equation.
-2.583333334 + 2.583333334 + x = 3.465103815 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + x = 3.465103815 + 2.583333334
x = 3.465103815 + 2.583333334

Combine like terms: 3.465103815 + 2.583333334 = 6.048437149
x = 6.048437149

Simplifying
x = 6.048437149


Subproblem 2
x + -2.583333334 = -3.465103815

Simplifying
x + -2.583333334 = -3.465103815

Reorder the terms:
-2.583333334 + x = -3.465103815

Solving
-2.583333334 + x = -3.465103815

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '2.583333334' to each side of the equation.
-2.583333334 + 2.583333334 + x = -3.465103815 + 2.583333334

Combine like terms: -2.583333334 + 2.583333334 = 0.000000000
0.000000000 + x = -3.465103815 + 2.583333334
x = -3.465103815 + 2.583333334

Combine like terms: -3.465103815 + 2.583333334 = -0.881770481
x = -0.881770481

Simplifying
x = -0.881770481


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.048437149, -0.881770481}

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