8n^2-16n-41=0

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Solution for 8n^2-16n-41=0 equation:


Simplifying
8n2 + -16n + -41 = 0

Reorder the terms:
-41 + -16n + 8n2 = 0

Solving
-41 + -16n + 8n2 = 0

Solving for variable 'n'.

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

Divide each side by '8'.
-5.125 + -2n + n2 = 0

Move the constant term to the right:

Add '5.125' to each side of the equation.
-5.125 + -2n + 5.125 + n2 = 0 + 5.125

Reorder the terms:
-5.125 + 5.125 + -2n + n2 = 0 + 5.125

Combine like terms: -5.125 + 5.125 = 0.000
0.000 + -2n + n2 = 0 + 5.125
-2n + n2 = 0 + 5.125

Combine like terms: 0 + 5.125 = 5.125
-2n + n2 = 5.125

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 5.125 + 1

Reorder the terms:
1 + -2n + n2 = 5.125 + 1

Combine like terms: 5.125 + 1 = 6.125
1 + -2n + n2 = 6.125

Factor a perfect square on the left side:
(n + -1)(n + -1) = 6.125

Calculate the square root of the right side: 2.474873734

Break this problem into two subproblems by setting 
(n + -1) equal to 2.474873734 and -2.474873734.

Subproblem 1

n + -1 = 2.474873734 Simplifying n + -1 = 2.474873734 Reorder the terms: -1 + n = 2.474873734 Solving -1 + n = 2.474873734 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 2.474873734 + 1 Combine like terms: -1 + 1 = 0 0 + n = 2.474873734 + 1 n = 2.474873734 + 1 Combine like terms: 2.474873734 + 1 = 3.474873734 n = 3.474873734 Simplifying n = 3.474873734

Subproblem 2

n + -1 = -2.474873734 Simplifying n + -1 = -2.474873734 Reorder the terms: -1 + n = -2.474873734 Solving -1 + n = -2.474873734 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -2.474873734 + 1 Combine like terms: -1 + 1 = 0 0 + n = -2.474873734 + 1 n = -2.474873734 + 1 Combine like terms: -2.474873734 + 1 = -1.474873734 n = -1.474873734 Simplifying n = -1.474873734

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.474873734, -1.474873734}

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