n^2=8n+4

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


Simplifying
n2 = 8n + 4

Reorder the terms:
n2 = 4 + 8n

Solving
n2 = 4 + 8n

Solving for variable 'n'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

The n term is -8n.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8n + 16 + n2 = 4 + 16

Reorder the terms:
16 + -8n + n2 = 4 + 16

Combine like terms: 4 + 16 = 20
16 + -8n + n2 = 20

Factor a perfect square on the left side:
(n + -4)(n + -4) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(n + -4) equal to 4.472135955 and -4.472135955.

Subproblem 1

n + -4 = 4.472135955 Simplifying n + -4 = 4.472135955 Reorder the terms: -4 + n = 4.472135955 Solving -4 + n = 4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 4.472135955 + 4 Combine like terms: -4 + 4 = 0 0 + n = 4.472135955 + 4 n = 4.472135955 + 4 Combine like terms: 4.472135955 + 4 = 8.472135955 n = 8.472135955 Simplifying n = 8.472135955

Subproblem 2

n + -4 = -4.472135955 Simplifying n + -4 = -4.472135955 Reorder the terms: -4 + n = -4.472135955 Solving -4 + n = -4.472135955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -4.472135955 + 4 Combine like terms: -4 + 4 = 0 0 + n = -4.472135955 + 4 n = -4.472135955 + 4 Combine like terms: -4.472135955 + 4 = -0.472135955 n = -0.472135955 Simplifying n = -0.472135955

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.472135955, -0.472135955}

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