8n^2+4n=1

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


Simplifying
8n2 + 4n = 1

Reorder the terms:
4n + 8n2 = 1

Solving
4n + 8n2 = 1

Solving for variable 'n'.

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

Combine like terms: 1 + -1 = 0
-1 + 4n + 8n2 = 0

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

Divide each side by '8'.
-0.125 + 0.5n + n2 = 0

Move the constant term to the right:

Add '0.125' to each side of the equation.
-0.125 + 0.5n + 0.125 + n2 = 0 + 0.125

Reorder the terms:
-0.125 + 0.125 + 0.5n + n2 = 0 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + 0.5n + n2 = 0 + 0.125
0.5n + n2 = 0 + 0.125

Combine like terms: 0 + 0.125 = 0.125
0.5n + n2 = 0.125

The n term is 0.5n.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5n + 0.0625 + n2 = 0.125 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 0.125 + 0.0625

Combine like terms: 0.125 + 0.0625 = 0.1875
0.0625 + 0.5n + n2 = 0.1875

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 0.1875

Calculate the square root of the right side: 0.433012702

Break this problem into two subproblems by setting 
(n + 0.25) equal to 0.433012702 and -0.433012702.

Subproblem 1

n + 0.25 = 0.433012702 Simplifying n + 0.25 = 0.433012702 Reorder the terms: 0.25 + n = 0.433012702 Solving 0.25 + n = 0.433012702 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 0.433012702 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 0.433012702 + -0.25 n = 0.433012702 + -0.25 Combine like terms: 0.433012702 + -0.25 = 0.183012702 n = 0.183012702 Simplifying n = 0.183012702

Subproblem 2

n + 0.25 = -0.433012702 Simplifying n + 0.25 = -0.433012702 Reorder the terms: 0.25 + n = -0.433012702 Solving 0.25 + n = -0.433012702 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -0.433012702 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -0.433012702 + -0.25 n = -0.433012702 + -0.25 Combine like terms: -0.433012702 + -0.25 = -0.683012702 n = -0.683012702 Simplifying n = -0.683012702

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.183012702, -0.683012702}

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