2x(8x+3)=46

Solution for 2x(8x+3)=46 equation:

Simplifying
2x(8x + 3) = 46

Reorder the terms:
2x(3 + 8x) = 46
(3 * 2x + 8x * 2x) = 46
(6x + 16x2) = 46

Solving
6x + 16x2 = 46

Solving for variable 'x'.

Reorder the terms:
-46 + 6x + 16x2 = 46 + -46

Combine like terms: 46 + -46 = 0
-46 + 6x + 16x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-23 + 3x + 8x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-23 + 3x + 8x2)' equal to zero and attempt to solve:

Simplifying
-23 + 3x + 8x2 = 0

Solving
-23 + 3x + 8x2 = 0

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

Divide each side by '8'.
-2.875 + 0.375x + x2 = 0

Move the constant term to the right:

Add '2.875' to each side of the equation.
-2.875 + 0.375x + 2.875 + x2 = 0 + 2.875

Reorder the terms:
-2.875 + 2.875 + 0.375x + x2 = 0 + 2.875

Combine like terms: -2.875 + 2.875 = 0.000
0.000 + 0.375x + x2 = 0 + 2.875
0.375x + x2 = 0 + 2.875

Combine like terms: 0 + 2.875 = 2.875
0.375x + x2 = 2.875

The x term is 0.375x.  Take half its coefficient (0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
0.375x + 0.03515625 + x2 = 2.875 + 0.03515625

Reorder the terms:
0.03515625 + 0.375x + x2 = 2.875 + 0.03515625

Combine like terms: 2.875 + 0.03515625 = 2.91015625
0.03515625 + 0.375x + x2 = 2.91015625

Factor a perfect square on the left side:
(x + 0.1875)(x + 0.1875) = 2.91015625

Calculate the square root of the right side: 1.705918008

Break this problem into two subproblems by setting 
(x + 0.1875) equal to 1.705918008 and -1.705918008.

Subproblem 1
x + 0.1875 = 1.705918008

Simplifying
x + 0.1875 = 1.705918008

Reorder the terms:
0.1875 + x = 1.705918008

Solving
0.1875 + x = 1.705918008

Solving for variable 'x'.

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

Add '-0.1875' to each side of the equation.
0.1875 + -0.1875 + x = 1.705918008 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + x = 1.705918008 + -0.1875
x = 1.705918008 + -0.1875

Combine like terms: 1.705918008 + -0.1875 = 1.518418008
x = 1.518418008

Simplifying
x = 1.518418008

Subproblem 2
x + 0.1875 = -1.705918008

Simplifying
x + 0.1875 = -1.705918008

Reorder the terms:
0.1875 + x = -1.705918008

Solving
0.1875 + x = -1.705918008

Solving for variable 'x'.

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

Add '-0.1875' to each side of the equation.
0.1875 + -0.1875 + x = -1.705918008 + -0.1875

Combine like terms: 0.1875 + -0.1875 = 0.0000
0.0000 + x = -1.705918008 + -0.1875
x = -1.705918008 + -0.1875

Combine like terms: -1.705918008 + -0.1875 = -1.893418008
x = -1.893418008

Simplifying
x = -1.893418008

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.518418008, -1.893418008}
Solution
x = {1.518418008, -1.893418008}