11n^2+2n=1

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Solution for 11n^2+2n=1 equation:


Simplifying
11n2 + 2n = 1

Reorder the terms:
2n + 11n2 = 1

Solving
2n + 11n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + 2n + 11n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 2n + 11n2 = 0

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

Divide each side by '11'.
-0.09090909091 + 0.1818181818n + n2 = 0

Move the constant term to the right:

Add '0.09090909091' to each side of the equation.
-0.09090909091 + 0.1818181818n + 0.09090909091 + n2 = 0 + 0.09090909091

Reorder the terms:
-0.09090909091 + 0.09090909091 + 0.1818181818n + n2 = 0 + 0.09090909091

Combine like terms: -0.09090909091 + 0.09090909091 = 0.00000000000
0.00000000000 + 0.1818181818n + n2 = 0 + 0.09090909091
0.1818181818n + n2 = 0 + 0.09090909091

Combine like terms: 0 + 0.09090909091 = 0.09090909091
0.1818181818n + n2 = 0.09090909091

The n term is 0.1818181818n.  Take half its coefficient (0.0909090909).
Square it (0.008264462808) and add it to both sides.

Add '0.008264462808' to each side of the equation.
0.1818181818n + 0.008264462808 + n2 = 0.09090909091 + 0.008264462808

Reorder the terms:
0.008264462808 + 0.1818181818n + n2 = 0.09090909091 + 0.008264462808

Combine like terms: 0.09090909091 + 0.008264462808 = 0.099173553718
0.008264462808 + 0.1818181818n + n2 = 0.099173553718

Factor a perfect square on the left side:
(n + 0.0909090909)(n + 0.0909090909) = 0.099173553718

Calculate the square root of the right side: 0.314918329

Break this problem into two subproblems by setting 
(n + 0.0909090909) equal to 0.314918329 and -0.314918329.

Subproblem 1

n + 0.0909090909 = 0.314918329 Simplifying n + 0.0909090909 = 0.314918329 Reorder the terms: 0.0909090909 + n = 0.314918329 Solving 0.0909090909 + n = 0.314918329 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + n = 0.314918329 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + n = 0.314918329 + -0.0909090909 n = 0.314918329 + -0.0909090909 Combine like terms: 0.314918329 + -0.0909090909 = 0.2240092381 n = 0.2240092381 Simplifying n = 0.2240092381

Subproblem 2

n + 0.0909090909 = -0.314918329 Simplifying n + 0.0909090909 = -0.314918329 Reorder the terms: 0.0909090909 + n = -0.314918329 Solving 0.0909090909 + n = -0.314918329 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.0909090909' to each side of the equation. 0.0909090909 + -0.0909090909 + n = -0.314918329 + -0.0909090909 Combine like terms: 0.0909090909 + -0.0909090909 = 0.0000000000 0.0000000000 + n = -0.314918329 + -0.0909090909 n = -0.314918329 + -0.0909090909 Combine like terms: -0.314918329 + -0.0909090909 = -0.4058274199 n = -0.4058274199 Simplifying n = -0.4058274199

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.2240092381, -0.4058274199}

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