3(1+2E)E=7

Solution for 3(1+2E)E=7 equation:

Simplifying
3(1 + 2E) * E = 7

Reorder the terms for easier multiplication:
3E(1 + 2E) = 7
(1 * 3E + 2E * 3E) = 7
(3E + 6E2) = 7

Solving
3E + 6E2 = 7

Solving for variable 'E'.

Reorder the terms:
-7 + 3E + 6E2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 3E + 6E2 = 0

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

Divide each side by '6'.
-1.166666667 + 0.5E + E2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + 0.5E + 1.166666667 + E2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + 0.5E + E2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + 0.5E + E2 = 0 + 1.166666667
0.5E + E2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
0.5E + E2 = 1.166666667

The E term is 0.5E.  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.
0.5E + 0.0625 + E2 = 1.166666667 + 0.0625

Reorder the terms:
0.0625 + 0.5E + E2 = 1.166666667 + 0.0625

Combine like terms: 1.166666667 + 0.0625 = 1.229166667
0.0625 + 0.5E + E2 = 1.229166667

Factor a perfect square on the left side:
(E + 0.25)(E + 0.25) = 1.229166667

Calculate the square root of the right side: 1.108677891

Break this problem into two subproblems by setting 
(E + 0.25) equal to 1.108677891 and -1.108677891.

Subproblem 1
E + 0.25 = 1.108677891

Simplifying
E + 0.25 = 1.108677891

Reorder the terms:
0.25 + E = 1.108677891

Solving
0.25 + E = 1.108677891

Solving for variable 'E'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + E = 1.108677891 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + E = 1.108677891 + -0.25
E = 1.108677891 + -0.25

Combine like terms: 1.108677891 + -0.25 = 0.858677891
E = 0.858677891

Simplifying
E = 0.858677891

Subproblem 2
E + 0.25 = -1.108677891

Simplifying
E + 0.25 = -1.108677891

Reorder the terms:
0.25 + E = -1.108677891

Solving
0.25 + E = -1.108677891

Solving for variable 'E'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + E = -1.108677891 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + E = -1.108677891 + -0.25
E = -1.108677891 + -0.25

Combine like terms: -1.108677891 + -0.25 = -1.358677891
E = -1.358677891

Simplifying
E = -1.358677891

Solution
The solution to the problem is based on the solutions
from the subproblems.
E = {0.858677891, -1.358677891}