4n^2-8n+1=0

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Solution for 4n^2-8n+1=0 equation:


Simplifying
4n2 + -8n + 1 = 0

Reorder the terms:
1 + -8n + 4n2 = 0

Solving
1 + -8n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
0.25 + -2n + n2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -2n + -0.25 + n2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -2n + n2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -2n + n2 = 0 + -0.25
-2n + n2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-2n + n2 = -0.25

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 = -0.25 + 1

Reorder the terms:
1 + -2n + n2 = -0.25 + 1

Combine like terms: -0.25 + 1 = 0.75
1 + -2n + n2 = 0.75

Factor a perfect square on the left side:
(n + -1)(n + -1) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(n + -1) equal to 0.866025404 and -0.866025404.

Subproblem 1

n + -1 = 0.866025404 Simplifying n + -1 = 0.866025404 Reorder the terms: -1 + n = 0.866025404 Solving -1 + n = 0.866025404 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 = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + n = 0.866025404 + 1 n = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 n = 1.866025404 Simplifying n = 1.866025404

Subproblem 2

n + -1 = -0.866025404 Simplifying n + -1 = -0.866025404 Reorder the terms: -1 + n = -0.866025404 Solving -1 + n = -0.866025404 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 = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + n = -0.866025404 + 1 n = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 n = 0.133974596 Simplifying n = 0.133974596

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.866025404, 0.133974596}

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