e^2+7e=-4

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Solution for e^2+7e=-4 equation: 


Simplifying
e2 + 7e = -4

Reorder the terms:
7e + e2 = -4

Solving
7e + e2 = -4

Solving for variable 'e'.

Reorder the terms:
4 + 7e + e2 = -4 + 4

Combine like terms: -4 + 4 = 0
4 + 7e + e2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 7e + -4 + e2 = 0 + -4

Reorder the terms:
4 + -4 + 7e + e2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 7e + e2 = 0 + -4
7e + e2 = 0 + -4

Combine like terms: 0 + -4 = -4
7e + e2 = -4

The e term is 7e.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7e + 12.25 + e2 = -4 + 12.25

Reorder the terms:
12.25 + 7e + e2 = -4 + 12.25

Combine like terms: -4 + 12.25 = 8.25
12.25 + 7e + e2 = 8.25

Factor a perfect square on the left side:
(e + 3.5)(e + 3.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(e + 3.5) equal to 2.872281323 and -2.872281323.


Subproblem 1
e + 3.5 = 2.872281323

Simplifying
e + 3.5 = 2.872281323

Reorder the terms:
3.5 + e = 2.872281323

Solving
3.5 + e = 2.872281323

Solving for variable 'e'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + e = 2.872281323 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + e = 2.872281323 + -3.5
e = 2.872281323 + -3.5

Combine like terms: 2.872281323 + -3.5 = -0.627718677
e = -0.627718677

Simplifying
e = -0.627718677


Subproblem 2
e + 3.5 = -2.872281323

Simplifying
e + 3.5 = -2.872281323

Reorder the terms:
3.5 + e = -2.872281323

Solving
3.5 + e = -2.872281323

Solving for variable 'e'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + e = -2.872281323 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + e = -2.872281323 + -3.5
e = -2.872281323 + -3.5

Combine like terms: -2.872281323 + -3.5 = -6.372281323
e = -6.372281323

Simplifying
e = -6.372281323


Solution
The solution to the problem is based on the solutions
from the subproblems.
e = {-0.627718677, -6.372281323}

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