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

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

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

Reorder the terms:
x(-7 + x2) = (x2 + 1)(x + -3)
(-7 * x + x2 * x) = (x2 + 1)(x + -3)
(-7x + x3) = (x2 + 1)(x + -3)

Reorder the terms:
-7x + x3 = (1 + x2)(x + -3)

Reorder the terms:
-7x + x3 = (1 + x2)(-3 + x)

Multiply (1 + x2) * (-3 + x)
-7x + x3 = (1(-3 + x) + x2(-3 + x))
-7x + x3 = ((-3 * 1 + x * 1) + x2(-3 + x))
-7x + x3 = ((-3 + 1x) + x2(-3 + x))
-7x + x3 = (-3 + 1x + (-3 * x2 + x * x2))
-7x + x3 = (-3 + 1x + (-3x2 + x3))
-7x + x3 = (-3 + 1x + -3x2 + x3)

Add '-1x3' to each side of the equation.
-7x + x3 + -1x3 = -3 + 1x + -3x2 + x3 + -1x3

Combine like terms: x3 + -1x3 = 0
-7x + 0 = -3 + 1x + -3x2 + x3 + -1x3
-7x = -3 + 1x + -3x2 + x3 + -1x3

Combine like terms: x3 + -1x3 = 0
-7x = -3 + 1x + -3x2 + 0
-7x = -3 + 1x + -3x2

Solving
-7x = -3 + 1x + -3x2

Solving for variable 'x'.

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

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

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

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

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

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

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

Divide each side by '3'.
1 + -2.666666667x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -2.666666667x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -2.666666667x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -2.666666667x + x2 = 0 + -1
-2.666666667x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-2.666666667x + x2 = -1

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 = -1 + 1.777777780

Reorder the terms:
1.777777780 + -2.666666667x + x2 = -1 + 1.777777780

Combine like terms: -1 + 1.777777780 = 0.77777778
1.777777780 + -2.666666667x + x2 = 0.77777778

Factor a perfect square on the left side:
(x + -1.333333334)(x + -1.333333334) = 0.77777778

Calculate the square root of the right side: 0.881917105

Break this problem into two subproblems by setting 
(x + -1.333333334) equal to 0.881917105 and -0.881917105.

Subproblem 1
x + -1.333333334 = 0.881917105

Simplifying
x + -1.333333334 = 0.881917105

Reorder the terms:
-1.333333334 + x = 0.881917105

Solving
-1.333333334 + x = 0.881917105

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 = 0.881917105 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = 0.881917105 + 1.333333334
x = 0.881917105 + 1.333333334

Combine like terms: 0.881917105 + 1.333333334 = 2.215250439
x = 2.215250439

Simplifying
x = 2.215250439

Subproblem 2
x + -1.333333334 = -0.881917105

Simplifying
x + -1.333333334 = -0.881917105

Reorder the terms:
-1.333333334 + x = -0.881917105

Solving
-1.333333334 + x = -0.881917105

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 = -0.881917105 + 1.333333334

Combine like terms: -1.333333334 + 1.333333334 = 0.000000000
0.000000000 + x = -0.881917105 + 1.333333334
x = -0.881917105 + 1.333333334

Combine like terms: -0.881917105 + 1.333333334 = 0.451416229
x = 0.451416229

Simplifying
x = 0.451416229

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.215250439, 0.451416229}