4e+3=2e^2

Solution for 4e+3=2e^2 equation:

Simplifying
4e + 3 = 2e2

Reorder the terms:
3 + 4e = 2e2

Solving
3 + 4e = 2e2

Solving for variable 'e'.

Combine like terms: 2e2 + -2e2 = 0
3 + 4e + -2e2 = 0

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

Divide each side by '-2'.
-1.5 + -2e + e2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -2e + 1.5 + e2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -2e + e2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -2e + e2 = 0 + 1.5
-2e + e2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-2e + e2 = 1.5

The e term is -2e.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2e + 1 + e2 = 1.5 + 1

Reorder the terms:
1 + -2e + e2 = 1.5 + 1

Combine like terms: 1.5 + 1 = 2.5
1 + -2e + e2 = 2.5

Factor a perfect square on the left side:
(e + -1)(e + -1) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(e + -1) equal to 1.58113883 and -1.58113883.

Subproblem 1
e + -1 = 1.58113883

Simplifying
e + -1 = 1.58113883

Reorder the terms:
-1 + e = 1.58113883

Solving
-1 + e = 1.58113883

Solving for variable 'e'.

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

Add '1' to each side of the equation.
-1 + 1 + e = 1.58113883 + 1

Combine like terms: -1 + 1 = 0
0 + e = 1.58113883 + 1
e = 1.58113883 + 1

Combine like terms: 1.58113883 + 1 = 2.58113883
e = 2.58113883

Simplifying
e = 2.58113883

Subproblem 2
e + -1 = -1.58113883

Simplifying
e + -1 = -1.58113883

Reorder the terms:
-1 + e = -1.58113883

Solving
-1 + e = -1.58113883

Solving for variable 'e'.

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

Add '1' to each side of the equation.
-1 + 1 + e = -1.58113883 + 1

Combine like terms: -1 + 1 = 0
0 + e = -1.58113883 + 1
e = -1.58113883 + 1

Combine like terms: -1.58113883 + 1 = -0.58113883
e = -0.58113883

Simplifying
e = -0.58113883

Solution
The solution to the problem is based on the solutions
from the subproblems.
e = {2.58113883, -0.58113883}