8n+8=n^2+4n

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Solution for 8n+8=n^2+4n equation:


Simplifying
8n + 8 = n2 + 4n

Reorder the terms:
8 + 8n = n2 + 4n

Reorder the terms:
8 + 8n = 4n + n2

Solving
8 + 8n = 4n + n2

Solving for variable 'n'.

Combine like terms: 8n + -4n = 4n
8 + 4n + -1n2 = 4n + n2 + -4n + -1n2

Reorder the terms:
8 + 4n + -1n2 = 4n + -4n + n2 + -1n2

Combine like terms: 4n + -4n = 0
8 + 4n + -1n2 = 0 + n2 + -1n2
8 + 4n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
8 + 4n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-8 + -4n + n2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -4n + 8 + n2 = 0 + 8

Reorder the terms:
-8 + 8 + -4n + n2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -4n + n2 = 0 + 8
-4n + n2 = 0 + 8

Combine like terms: 0 + 8 = 8
-4n + n2 = 8

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 8 + 4

Reorder the terms:
4 + -4n + n2 = 8 + 4

Combine like terms: 8 + 4 = 12
4 + -4n + n2 = 12

Factor a perfect square on the left side:
(n + -2)(n + -2) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(n + -2) equal to 3.464101615 and -3.464101615.

Subproblem 1

n + -2 = 3.464101615 Simplifying n + -2 = 3.464101615 Reorder the terms: -2 + n = 3.464101615 Solving -2 + n = 3.464101615 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 3.464101615 + 2 Combine like terms: -2 + 2 = 0 0 + n = 3.464101615 + 2 n = 3.464101615 + 2 Combine like terms: 3.464101615 + 2 = 5.464101615 n = 5.464101615 Simplifying n = 5.464101615

Subproblem 2

n + -2 = -3.464101615 Simplifying n + -2 = -3.464101615 Reorder the terms: -2 + n = -3.464101615 Solving -2 + n = -3.464101615 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -3.464101615 + 2 Combine like terms: -2 + 2 = 0 0 + n = -3.464101615 + 2 n = -3.464101615 + 2 Combine like terms: -3.464101615 + 2 = -1.464101615 n = -1.464101615 Simplifying n = -1.464101615

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.464101615, -1.464101615}

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