6x^2-3x-8=0

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Solution for 6x^2-3x-8=0 equation: 


Simplifying
6x2 + -3x + -8 = 0

Reorder the terms:
-8 + -3x + 6x2 = 0

Solving
-8 + -3x + 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'.
-1.333333333 + -0.5x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -0.5x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -0.5x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -0.5x + x2 = 0 + 1.333333333
-0.5x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-0.5x + x2 = 1.333333333

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 = 1.333333333 + 0.0625

Reorder the terms:
0.0625 + -0.5x + x2 = 1.333333333 + 0.0625

Combine like terms: 1.333333333 + 0.0625 = 1.395833333
0.0625 + -0.5x + x2 = 1.395833333

Factor a perfect square on the left side:
(x + -0.25)(x + -0.25) = 1.395833333

Calculate the square root of the right side: 1.181453906

Break this problem into two subproblems by setting 
(x + -0.25) equal to 1.181453906 and -1.181453906.


Subproblem 1
x + -0.25 = 1.181453906

Simplifying
x + -0.25 = 1.181453906

Reorder the terms:
-0.25 + x = 1.181453906

Solving
-0.25 + x = 1.181453906

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 = 1.181453906 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = 1.181453906 + 0.25
x = 1.181453906 + 0.25

Combine like terms: 1.181453906 + 0.25 = 1.431453906
x = 1.431453906

Simplifying
x = 1.431453906


Subproblem 2
x + -0.25 = -1.181453906

Simplifying
x + -0.25 = -1.181453906

Reorder the terms:
-0.25 + x = -1.181453906

Solving
-0.25 + x = -1.181453906

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 = -1.181453906 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + x = -1.181453906 + 0.25
x = -1.181453906 + 0.25

Combine like terms: -1.181453906 + 0.25 = -0.931453906
x = -0.931453906

Simplifying
x = -0.931453906


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.431453906, -0.931453906}

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