6n=n^2+2

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Solution for 6n=n^2+2 equation:


Simplifying
6n = n2 + 2

Reorder the terms:
6n = 2 + n2

Solving
6n = 2 + n2

Solving for variable 'n'.

Reorder the terms:
-2 + 6n + -1n2 = 2 + n2 + -2 + -1n2

Reorder the terms:
-2 + 6n + -1n2 = 2 + -2 + n2 + -1n2

Combine like terms: 2 + -2 = 0
-2 + 6n + -1n2 = 0 + n2 + -1n2
-2 + 6n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-2 + 6n + -1n2 = 0

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

Divide each side by '-1'.
2 + -6n + n2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -6n + -2 + n2 = 0 + -2

Reorder the terms:
2 + -2 + -6n + n2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -6n + n2 = 0 + -2
-6n + n2 = 0 + -2

Combine like terms: 0 + -2 = -2
-6n + n2 = -2

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = -2 + 9

Reorder the terms:
9 + -6n + n2 = -2 + 9

Combine like terms: -2 + 9 = 7
9 + -6n + n2 = 7

Factor a perfect square on the left side:
(n + -3)(n + -3) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(n + -3) equal to 2.645751311 and -2.645751311.

Subproblem 1

n + -3 = 2.645751311 Simplifying n + -3 = 2.645751311 Reorder the terms: -3 + n = 2.645751311 Solving -3 + n = 2.645751311 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 2.645751311 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.645751311 + 3 n = 2.645751311 + 3 Combine like terms: 2.645751311 + 3 = 5.645751311 n = 5.645751311 Simplifying n = 5.645751311

Subproblem 2

n + -3 = -2.645751311 Simplifying n + -3 = -2.645751311 Reorder the terms: -3 + n = -2.645751311 Solving -3 + n = -2.645751311 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -2.645751311 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.645751311 + 3 n = -2.645751311 + 3 Combine like terms: -2.645751311 + 3 = 0.354248689 n = 0.354248689 Simplifying n = 0.354248689

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.645751311, 0.354248689}

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