3x^2-11x-9=x^2+6x-x(x+1)+1

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Solution for 3x^2-11x-9=x^2+6x-x(x+1)+1 equation: 


Simplifying
3x2 + -11x + -9 = x2 + 6x + -1x(x + 1) + 1

Reorder the terms:
-9 + -11x + 3x2 = x2 + 6x + -1x(x + 1) + 1

Reorder the terms:
-9 + -11x + 3x2 = x2 + 6x + -1x(1 + x) + 1
-9 + -11x + 3x2 = x2 + 6x + (1 * -1x + x * -1x) + 1
-9 + -11x + 3x2 = x2 + 6x + (-1x + -1x2) + 1

Reorder the terms:
-9 + -11x + 3x2 = 1 + 6x + -1x + x2 + -1x2

Combine like terms: 6x + -1x = 5x
-9 + -11x + 3x2 = 1 + 5x + x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-9 + -11x + 3x2 = 1 + 5x + 0
-9 + -11x + 3x2 = 1 + 5x

Solving
-9 + -11x + 3x2 = 1 + 5x

Solving for variable 'x'.

Reorder the terms:
-9 + -1 + -11x + -5x + 3x2 = 1 + 5x + -1 + -5x

Combine like terms: -9 + -1 = -10
-10 + -11x + -5x + 3x2 = 1 + 5x + -1 + -5x

Combine like terms: -11x + -5x = -16x
-10 + -16x + 3x2 = 1 + 5x + -1 + -5x

Reorder the terms:
-10 + -16x + 3x2 = 1 + -1 + 5x + -5x

Combine like terms: 1 + -1 = 0
-10 + -16x + 3x2 = 0 + 5x + -5x
-10 + -16x + 3x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-10 + -16x + 3x2 = 0

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

Divide each side by '3'.
-3.333333333 + -5.333333333x + x2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -5.333333333x + 3.333333333 + x2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -5.333333333x + x2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -5.333333333x + x2 = 0 + 3.333333333
-5.333333333x + x2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-5.333333333x + x2 = 3.333333333

The x term is -5.333333333x.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333x + 7.111111113 + x2 = 3.333333333 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333x + x2 = 3.333333333 + 7.111111113

Combine like terms: 3.333333333 + 7.111111113 = 10.444444446
7.111111113 + -5.333333333x + x2 = 10.444444446

Factor a perfect square on the left side:
(x + -2.666666667)(x + -2.666666667) = 10.444444446

Calculate the square root of the right side: 3.231786572

Break this problem into two subproblems by setting 
(x + -2.666666667) equal to 3.231786572 and -3.231786572.


Subproblem 1
x + -2.666666667 = 3.231786572

Simplifying
x + -2.666666667 = 3.231786572

Reorder the terms:
-2.666666667 + x = 3.231786572

Solving
-2.666666667 + x = 3.231786572

Solving for variable 'x'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + x = 3.231786572 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + x = 3.231786572 + 2.666666667
x = 3.231786572 + 2.666666667

Combine like terms: 3.231786572 + 2.666666667 = 5.898453239
x = 5.898453239

Simplifying
x = 5.898453239


Subproblem 2
x + -2.666666667 = -3.231786572

Simplifying
x + -2.666666667 = -3.231786572

Reorder the terms:
-2.666666667 + x = -3.231786572

Solving
-2.666666667 + x = -3.231786572

Solving for variable 'x'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + x = -3.231786572 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + x = -3.231786572 + 2.666666667
x = -3.231786572 + 2.666666667

Combine like terms: -3.231786572 + 2.666666667 = -0.565119905
x = -0.565119905

Simplifying
x = -0.565119905


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.898453239, -0.565119905}

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