3x^2-8x-6=x^2+3x-x(x+1)+1

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Solution for 3x^2-8x-6=x^2+3x-x(x+1)+1 equation: 


Simplifying
3x2 + -8x + -6 = x2 + 3x + -1x(x + 1) + 1

Reorder the terms:
-6 + -8x + 3x2 = x2 + 3x + -1x(x + 1) + 1

Reorder the terms:
-6 + -8x + 3x2 = x2 + 3x + -1x(1 + x) + 1
-6 + -8x + 3x2 = x2 + 3x + (1 * -1x + x * -1x) + 1
-6 + -8x + 3x2 = x2 + 3x + (-1x + -1x2) + 1

Reorder the terms:
-6 + -8x + 3x2 = 1 + 3x + -1x + x2 + -1x2

Combine like terms: 3x + -1x = 2x
-6 + -8x + 3x2 = 1 + 2x + x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-6 + -8x + 3x2 = 1 + 2x + 0
-6 + -8x + 3x2 = 1 + 2x

Solving
-6 + -8x + 3x2 = 1 + 2x

Solving for variable 'x'.

Reorder the terms:
-6 + -1 + -8x + -2x + 3x2 = 1 + 2x + -1 + -2x

Combine like terms: -6 + -1 = -7
-7 + -8x + -2x + 3x2 = 1 + 2x + -1 + -2x

Combine like terms: -8x + -2x = -10x
-7 + -10x + 3x2 = 1 + 2x + -1 + -2x

Reorder the terms:
-7 + -10x + 3x2 = 1 + -1 + 2x + -2x

Combine like terms: 1 + -1 = 0
-7 + -10x + 3x2 = 0 + 2x + -2x
-7 + -10x + 3x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-7 + -10x + 3x2 = 0

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

Divide each side by '3'.
-2.333333333 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -3.333333333x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -3.333333333x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -3.333333333x + x2 = 0 + 2.333333333
-3.333333333x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-3.333333333x + x2 = 2.333333333

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = 2.333333333 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = 2.333333333 + 2.777777779

Combine like terms: 2.333333333 + 2.777777779 = 5.111111112
2.777777779 + -3.333333333x + x2 = 5.111111112

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 5.111111112

Calculate the square root of the right side: 2.260776661

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 2.260776661 and -2.260776661.


Subproblem 1
x + -1.666666667 = 2.260776661

Simplifying
x + -1.666666667 = 2.260776661

Reorder the terms:
-1.666666667 + x = 2.260776661

Solving
-1.666666667 + x = 2.260776661

Solving for variable 'x'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = 2.260776661 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = 2.260776661 + 1.666666667
x = 2.260776661 + 1.666666667

Combine like terms: 2.260776661 + 1.666666667 = 3.927443328
x = 3.927443328

Simplifying
x = 3.927443328


Subproblem 2
x + -1.666666667 = -2.260776661

Simplifying
x + -1.666666667 = -2.260776661

Reorder the terms:
-1.666666667 + x = -2.260776661

Solving
-1.666666667 + x = -2.260776661

Solving for variable 'x'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + x = -2.260776661 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + x = -2.260776661 + 1.666666667
x = -2.260776661 + 1.666666667

Combine like terms: -2.260776661 + 1.666666667 = -0.594109994
x = -0.594109994

Simplifying
x = -0.594109994


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.927443328, -0.594109994}

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