6+2x^2-3x=8x^2

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

Simplifying
6 + 2x2 + -3x = 8x2

Reorder the terms:
6 + -3x + 2x2 = 8x2

Solving
6 + -3x + 2x2 = 8x2

Solving for variable 'x'.

Combine like terms: 2x2 + -8x2 = -6x2
6 + -3x + -6x2 = 8x2 + -8x2

Combine like terms: 8x2 + -8x2 = 0
6 + -3x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(2 + -1x + -2x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(2 + -1x + -2x2)' equal to zero and attempt to solve:

Simplifying
2 + -1x + -2x2 = 0

Solving
2 + -1x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-1 + 0.5x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.5x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.5x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.5x + x2 = 0 + 1
0.5x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.5x + x2 = 1

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 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 1 + 0.0625

Combine like terms: 1 + 0.0625 = 1.0625
0.0625 + 0.5x + x2 = 1.0625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(x + 0.25) equal to 1.030776406 and -1.030776406.

Subproblem 1
x + 0.25 = 1.030776406

Simplifying
x + 0.25 = 1.030776406

Reorder the terms:
0.25 + x = 1.030776406

Solving
0.25 + x = 1.030776406

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 = 1.030776406 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 1.030776406 + -0.25
x = 1.030776406 + -0.25

Combine like terms: 1.030776406 + -0.25 = 0.780776406
x = 0.780776406

Simplifying
x = 0.780776406

Subproblem 2
x + 0.25 = -1.030776406

Simplifying
x + 0.25 = -1.030776406

Reorder the terms:
0.25 + x = -1.030776406

Solving
0.25 + x = -1.030776406

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 = -1.030776406 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -1.030776406 + -0.25
x = -1.030776406 + -0.25

Combine like terms: -1.030776406 + -0.25 = -1.280776406
x = -1.280776406

Simplifying
x = -1.280776406

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.780776406, -1.280776406}
Solution
x = {0.780776406, -1.280776406}