8+2n=2n(2n+1)-n

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Solution for 8+2n=2n(2n+1)-n equation:


Simplifying
8 + 2n = 2n(2n + 1) + -1n

Reorder the terms:
8 + 2n = 2n(1 + 2n) + -1n
8 + 2n = (1 * 2n + 2n * 2n) + -1n
8 + 2n = (2n + 4n2) + -1n

Reorder the terms:
8 + 2n = 2n + -1n + 4n2

Combine like terms: 2n + -1n = 1n
8 + 2n = 1n + 4n2

Solving
8 + 2n = 1n + 4n2

Solving for variable 'n'.

Combine like terms: 2n + -1n = 1n
8 + 1n + -4n2 = 1n + 4n2 + -1n + -4n2

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

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

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

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-2 + -0.25n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -0.25n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + -0.25n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -0.25n + n2 = 0 + 2
-0.25n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
-0.25n + n2 = 2

The n term is -0.25n.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25n + 0.015625 + n2 = 2 + 0.015625

Reorder the terms:
0.015625 + -0.25n + n2 = 2 + 0.015625

Combine like terms: 2 + 0.015625 = 2.015625
0.015625 + -0.25n + n2 = 2.015625

Factor a perfect square on the left side:
(n + -0.125)(n + -0.125) = 2.015625

Calculate the square root of the right side: 1.419727086

Break this problem into two subproblems by setting 
(n + -0.125) equal to 1.419727086 and -1.419727086.

Subproblem 1

n + -0.125 = 1.419727086 Simplifying n + -0.125 = 1.419727086 Reorder the terms: -0.125 + n = 1.419727086 Solving -0.125 + n = 1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = 1.419727086 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 1.419727086 + 0.125 n = 1.419727086 + 0.125 Combine like terms: 1.419727086 + 0.125 = 1.544727086 n = 1.544727086 Simplifying n = 1.544727086

Subproblem 2

n + -0.125 = -1.419727086 Simplifying n + -0.125 = -1.419727086 Reorder the terms: -0.125 + n = -1.419727086 Solving -0.125 + n = -1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = -1.419727086 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -1.419727086 + 0.125 n = -1.419727086 + 0.125 Combine like terms: -1.419727086 + 0.125 = -1.294727086 n = -1.294727086 Simplifying n = -1.294727086

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.544727086, -1.294727086}

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