n^2-8n+1=0

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Solution for n^2-8n+1=0 equation:


Simplifying
n2 + -8n + 1 = 0

Reorder the terms:
1 + -8n + n2 = 0

Solving
1 + -8n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -8n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + -8n + n2 = 0 + -1

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

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

The n term is -8n.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8n + 16 + n2 = -1 + 16

Reorder the terms:
16 + -8n + n2 = -1 + 16

Combine like terms: -1 + 16 = 15
16 + -8n + n2 = 15

Factor a perfect square on the left side:
(n + -4)(n + -4) = 15

Calculate the square root of the right side: 3.872983346

Break this problem into two subproblems by setting 
(n + -4) equal to 3.872983346 and -3.872983346.

Subproblem 1

n + -4 = 3.872983346 Simplifying n + -4 = 3.872983346 Reorder the terms: -4 + n = 3.872983346 Solving -4 + n = 3.872983346 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = 3.872983346 + 4 Combine like terms: -4 + 4 = 0 0 + n = 3.872983346 + 4 n = 3.872983346 + 4 Combine like terms: 3.872983346 + 4 = 7.872983346 n = 7.872983346 Simplifying n = 7.872983346

Subproblem 2

n + -4 = -3.872983346 Simplifying n + -4 = -3.872983346 Reorder the terms: -4 + n = -3.872983346 Solving -4 + n = -3.872983346 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + n = -3.872983346 + 4 Combine like terms: -4 + 4 = 0 0 + n = -3.872983346 + 4 n = -3.872983346 + 4 Combine like terms: -3.872983346 + 4 = 0.127016654 n = 0.127016654 Simplifying n = 0.127016654

Solution

The solution to the problem is based on the solutions from the subproblems. n = {7.872983346, 0.127016654}

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