x(2x+1)=43

Solution for x(2x+1)=43 equation:

Simplifying
x(2x + 1) = 43

Reorder the terms:
x(1 + 2x) = 43
(1 * x + 2x * x) = 43
(1x + 2x2) = 43

Solving
1x + 2x2 = 43

Solving for variable 'x'.

Reorder the terms:
-43 + 1x + 2x2 = 43 + -43

Combine like terms: 43 + -43 = 0
-43 + 1x + 2x2 = 0

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

Divide each side by '2'.
-21.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '21.5' to each side of the equation.
-21.5 + 0.5x + 21.5 + x2 = 0 + 21.5

Reorder the terms:
-21.5 + 21.5 + 0.5x + x2 = 0 + 21.5

Combine like terms: -21.5 + 21.5 = 0.0
0.0 + 0.5x + x2 = 0 + 21.5
0.5x + x2 = 0 + 21.5

Combine like terms: 0 + 21.5 = 21.5
0.5x + x2 = 21.5

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 21.5 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 21.5 + 0.0625

Combine like terms: 21.5 + 0.0625 = 21.5625
0.0625 + 0.5x + x2 = 21.5625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 21.5625

Calculate the square root of the right side: 4.643543905

Break this problem into two subproblems by setting 
(x + 0.25) equal to 4.643543905 and -4.643543905.

Subproblem 1
x + 0.25 = 4.643543905

Simplifying
x + 0.25 = 4.643543905

Reorder the terms:
0.25 + x = 4.643543905

Solving
0.25 + x = 4.643543905

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = 4.643543905 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 4.643543905 + -0.25
x = 4.643543905 + -0.25

Combine like terms: 4.643543905 + -0.25 = 4.393543905
x = 4.393543905

Simplifying
x = 4.393543905

Subproblem 2
x + 0.25 = -4.643543905

Simplifying
x + 0.25 = -4.643543905

Reorder the terms:
0.25 + x = -4.643543905

Solving
0.25 + x = -4.643543905

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = -4.643543905 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -4.643543905 + -0.25
x = -4.643543905 + -0.25

Combine like terms: -4.643543905 + -0.25 = -4.893543905
x = -4.893543905

Simplifying
x = -4.893543905

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.393543905, -4.893543905}