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

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

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

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

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

Solving
-8x + x2 = 1 + 3x2

Solving for variable 'x'.

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

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

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

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

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

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 8x + 2x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(1 + 8x + 2x2)' equal to zero and attempt to solve:

Simplifying
1 + 8x + 2x2 = 0

Solving
1 + 8x + 2x2 = 0

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

Divide each side by '2'.
0.5 + 4x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 4x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 4x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 4x + x2 = 0 + -0.5
4x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
4x + x2 = -0.5

The x term is 4x.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4x + 4 + x2 = -0.5 + 4

Reorder the terms:
4 + 4x + x2 = -0.5 + 4

Combine like terms: -0.5 + 4 = 3.5
4 + 4x + x2 = 3.5

Factor a perfect square on the left side:
(x + 2)(x + 2) = 3.5

Calculate the square root of the right side: 1.870828693

Break this problem into two subproblems by setting 
(x + 2) equal to 1.870828693 and -1.870828693.

Subproblem 1
x + 2 = 1.870828693

Simplifying
x + 2 = 1.870828693

Reorder the terms:
2 + x = 1.870828693

Solving
2 + x = 1.870828693

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = 1.870828693 + -2

Combine like terms: 2 + -2 = 0
0 + x = 1.870828693 + -2
x = 1.870828693 + -2

Combine like terms: 1.870828693 + -2 = -0.129171307
x = -0.129171307

Simplifying
x = -0.129171307

Subproblem 2
x + 2 = -1.870828693

Simplifying
x + 2 = -1.870828693

Reorder the terms:
2 + x = -1.870828693

Solving
2 + x = -1.870828693

Solving for variable 'x'.

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

Add '-2' to each side of the equation.
2 + -2 + x = -1.870828693 + -2

Combine like terms: 2 + -2 = 0
0 + x = -1.870828693 + -2
x = -1.870828693 + -2

Combine like terms: -1.870828693 + -2 = -3.870828693
x = -3.870828693

Simplifying
x = -3.870828693

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.129171307, -3.870828693}
Solution
x = {-0.129171307, -3.870828693}