3x^2-6x-7=x^2+3x-x(x+1)+3

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Solution for 3x^2-6x-7=x^2+3x-x(x+1)+3 equation: 


Simplifying
3x2 + -6x + -7 = x2 + 3x + -1x(x + 1) + 3

Reorder the terms:
-7 + -6x + 3x2 = x2 + 3x + -1x(x + 1) + 3

Reorder the terms:
-7 + -6x + 3x2 = x2 + 3x + -1x(1 + x) + 3
-7 + -6x + 3x2 = x2 + 3x + (1 * -1x + x * -1x) + 3
-7 + -6x + 3x2 = x2 + 3x + (-1x + -1x2) + 3

Reorder the terms:
-7 + -6x + 3x2 = 3 + 3x + -1x + x2 + -1x2

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

Combine like terms: x2 + -1x2 = 0
-7 + -6x + 3x2 = 3 + 2x + 0
-7 + -6x + 3x2 = 3 + 2x

Solving
-7 + -6x + 3x2 = 3 + 2x

Solving for variable 'x'.

Reorder the terms:
-7 + -3 + -6x + -2x + 3x2 = 3 + 2x + -3 + -2x

Combine like terms: -7 + -3 = -10
-10 + -6x + -2x + 3x2 = 3 + 2x + -3 + -2x

Combine like terms: -6x + -2x = -8x
-10 + -8x + 3x2 = 3 + 2x + -3 + -2x

Reorder the terms:
-10 + -8x + 3x2 = 3 + -3 + 2x + -2x

Combine like terms: 3 + -3 = 0
-10 + -8x + 3x2 = 0 + 2x + -2x
-10 + -8x + 3x2 = 2x + -2x

Combine like terms: 2x + -2x = 0
-10 + -8x + 3x2 = 0

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

Divide each side by '3'.
-3.333333333 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -2.666666667x + 3.333333333 + x2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -2.666666667x + x2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -2.666666667x + x2 = 0 + 3.333333333
-2.666666667x + x2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-2.666666667x + x2 = 3.333333333

The x term is -2.666666667x.  Take half its coefficient (-1.333333334).
Square it (1.777777780) and add it to both sides.

Add '1.777777780' to each side of the equation.
-2.666666667x + 1.777777780 + x2 = 3.333333333 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = 3.333333333 + 1.777777780

Combine like terms: 3.333333333 + 1.777777780 = 5.111111113
1.777777780 + -2.666666667x + x2 = 5.111111113

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 5.111111113

Calculate the square root of the right side: 2.260776661

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 2.260776661 and -2.260776661.


Subproblem 1
x + -1.333333334 = 2.260776661

Simplifying
x + -1.333333334 = 2.260776661

Reorder the terms:
-1.333333334 + x = 2.260776661

Solving
-1.333333334 + x = 2.260776661

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = 2.260776661 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 2.260776661 + 1.333333334
x = 2.260776661 + 1.333333334

Combine like terms: 2.260776661 + 1.333333334 = 3.594109995
x = 3.594109995

Simplifying
x = 3.594109995


Subproblem 2
x + -1.333333334 = -2.260776661

Simplifying
x + -1.333333334 = -2.260776661

Reorder the terms:
-1.333333334 + x = -2.260776661

Solving
-1.333333334 + x = -2.260776661

Solving for variable 'x'.

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

Add '1.333333334' to each side of the equation.
-1.333333334 + 1.333333334 + x = -2.260776661 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -2.260776661 + 1.333333334
x = -2.260776661 + 1.333333334

Combine like terms: -2.260776661 + 1.333333334 = -0.927443327
x = -0.927443327

Simplifying
x = -0.927443327


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.594109995, -0.927443327}

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