5e(e+6)=4(e+7)

Solution for 5e(e+6)=4(e+7) equation:

Simplifying
5e(e + 6) = 4(e + 7)

Reorder the terms:
5e(6 + e) = 4(e + 7)
(6 * 5e + e * 5e) = 4(e + 7)
(30e + 5e2) = 4(e + 7)

Reorder the terms:
30e + 5e2 = 4(7 + e)
30e + 5e2 = (7 * 4 + e * 4)
30e + 5e2 = (28 + 4e)

Solving
30e + 5e2 = 28 + 4e

Solving for variable 'e'.

Reorder the terms:
-28 + 30e + -4e + 5e2 = 28 + 4e + -28 + -4e

Combine like terms: 30e + -4e = 26e
-28 + 26e + 5e2 = 28 + 4e + -28 + -4e

Reorder the terms:
-28 + 26e + 5e2 = 28 + -28 + 4e + -4e

Combine like terms: 28 + -28 = 0
-28 + 26e + 5e2 = 0 + 4e + -4e
-28 + 26e + 5e2 = 4e + -4e

Combine like terms: 4e + -4e = 0
-28 + 26e + 5e2 = 0

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

Divide each side by '5'.
-5.6 + 5.2e + e2 = 0

Move the constant term to the right:

Add '5.6' to each side of the equation.
-5.6 + 5.2e + 5.6 + e2 = 0 + 5.6

Reorder the terms:
-5.6 + 5.6 + 5.2e + e2 = 0 + 5.6

Combine like terms: -5.6 + 5.6 = 0.0
0.0 + 5.2e + e2 = 0 + 5.6
5.2e + e2 = 0 + 5.6

Combine like terms: 0 + 5.6 = 5.6
5.2e + e2 = 5.6

The e term is 5.2e.  Take half its coefficient (2.6).
Square it (6.76) and add it to both sides.

Add '6.76' to each side of the equation.
5.2e + 6.76 + e2 = 5.6 + 6.76

Reorder the terms:
6.76 + 5.2e + e2 = 5.6 + 6.76

Combine like terms: 5.6 + 6.76 = 12.36
6.76 + 5.2e + e2 = 12.36

Factor a perfect square on the left side:
(e + 2.6)(e + 2.6) = 12.36

Calculate the square root of the right side: 3.515679166

Break this problem into two subproblems by setting 
(e + 2.6) equal to 3.515679166 and -3.515679166.

Subproblem 1
e + 2.6 = 3.515679166

Simplifying
e + 2.6 = 3.515679166

Reorder the terms:
2.6 + e = 3.515679166

Solving
2.6 + e = 3.515679166

Solving for variable 'e'.

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

Add '-2.6' to each side of the equation.
2.6 + -2.6 + e = 3.515679166 + -2.6

Combine like terms: 2.6 + -2.6 = 0.0
0.0 + e = 3.515679166 + -2.6
e = 3.515679166 + -2.6

Combine like terms: 3.515679166 + -2.6 = 0.915679166
e = 0.915679166

Simplifying
e = 0.915679166

Subproblem 2
e + 2.6 = -3.515679166

Simplifying
e + 2.6 = -3.515679166

Reorder the terms:
2.6 + e = -3.515679166

Solving
2.6 + e = -3.515679166

Solving for variable 'e'.

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

Add '-2.6' to each side of the equation.
2.6 + -2.6 + e = -3.515679166 + -2.6

Combine like terms: 2.6 + -2.6 = 0.0
0.0 + e = -3.515679166 + -2.6
e = -3.515679166 + -2.6

Combine like terms: -3.515679166 + -2.6 = -6.115679166
e = -6.115679166

Simplifying
e = -6.115679166

Solution
The solution to the problem is based on the solutions
from the subproblems.
e = {0.915679166, -6.115679166}