8n^2-80n+193=0

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Solution for 8n^2-80n+193=0 equation:


Simplifying
8n2 + -80n + 193 = 0

Reorder the terms:
193 + -80n + 8n2 = 0

Solving
193 + -80n + 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'.
24.125 + -10n + n2 = 0

Move the constant term to the right:

Add '-24.125' to each side of the equation.
24.125 + -10n + -24.125 + n2 = 0 + -24.125

Reorder the terms:
24.125 + -24.125 + -10n + n2 = 0 + -24.125

Combine like terms: 24.125 + -24.125 = 0.000
0.000 + -10n + n2 = 0 + -24.125
-10n + n2 = 0 + -24.125

Combine like terms: 0 + -24.125 = -24.125
-10n + n2 = -24.125

The n term is -10n.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10n + 25 + n2 = -24.125 + 25

Reorder the terms:
25 + -10n + n2 = -24.125 + 25

Combine like terms: -24.125 + 25 = 0.875
25 + -10n + n2 = 0.875

Factor a perfect square on the left side:
(n + -5)(n + -5) = 0.875

Calculate the square root of the right side: 0.935414347

Break this problem into two subproblems by setting 
(n + -5) equal to 0.935414347 and -0.935414347.

Subproblem 1

n + -5 = 0.935414347 Simplifying n + -5 = 0.935414347 Reorder the terms: -5 + n = 0.935414347 Solving -5 + n = 0.935414347 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = 0.935414347 + 5 Combine like terms: -5 + 5 = 0 0 + n = 0.935414347 + 5 n = 0.935414347 + 5 Combine like terms: 0.935414347 + 5 = 5.935414347 n = 5.935414347 Simplifying n = 5.935414347

Subproblem 2

n + -5 = -0.935414347 Simplifying n + -5 = -0.935414347 Reorder the terms: -5 + n = -0.935414347 Solving -5 + n = -0.935414347 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + n = -0.935414347 + 5 Combine like terms: -5 + 5 = 0 0 + n = -0.935414347 + 5 n = -0.935414347 + 5 Combine like terms: -0.935414347 + 5 = 4.064585653 n = 4.064585653 Simplifying n = 4.064585653

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.935414347, 4.064585653}

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