6n-n^2=3

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Solution for 6n-n^2=3 equation:


Simplifying
6n + -1n2 = 3

Solving
6n + -1n2 = 3

Solving for variable 'n'.

Reorder the terms:
-3 + 6n + -1n2 = 3 + -3

Combine like terms: 3 + -3 = 0
-3 + 6n + -1n2 = 0

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

Divide each side by '-1'.
3 + -6n + n2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -6n + -3 + n2 = 0 + -3

Reorder the terms:
3 + -3 + -6n + n2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -6n + n2 = 0 + -3
-6n + n2 = 0 + -3

Combine like terms: 0 + -3 = -3
-6n + n2 = -3

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 = -3 + 9

Reorder the terms:
9 + -6n + n2 = -3 + 9

Combine like terms: -3 + 9 = 6
9 + -6n + n2 = 6

Factor a perfect square on the left side:
(n + -3)(n + -3) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(n + -3) equal to 2.449489743 and -2.449489743.

Subproblem 1

n + -3 = 2.449489743 Simplifying n + -3 = 2.449489743 Reorder the terms: -3 + n = 2.449489743 Solving -3 + n = 2.449489743 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 = 2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.449489743 + 3 n = 2.449489743 + 3 Combine like terms: 2.449489743 + 3 = 5.449489743 n = 5.449489743 Simplifying n = 5.449489743

Subproblem 2

n + -3 = -2.449489743 Simplifying n + -3 = -2.449489743 Reorder the terms: -3 + n = -2.449489743 Solving -3 + n = -2.449489743 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 = -2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.449489743 + 3 n = -2.449489743 + 3 Combine like terms: -2.449489743 + 3 = 0.550510257 n = 0.550510257 Simplifying n = 0.550510257

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.449489743, 0.550510257}

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