30n^2+n^2+n=1

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Solution for 30n^2+n^2+n=1 equation:


Simplifying
30n2 + n2 + n = 1

Reorder the terms:
n + 30n2 + n2 = 1

Combine like terms: 30n2 + n2 = 31n2
n + 31n2 = 1

Solving
n + 31n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + n + 31n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + n + 31n2 = 0

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

Divide each side by '31'.
-0.03225806452 + 0.03225806452n + n2 = 0

Move the constant term to the right:

Add '0.03225806452' to each side of the equation.
-0.03225806452 + 0.03225806452n + 0.03225806452 + n2 = 0 + 0.03225806452

Reorder the terms:
-0.03225806452 + 0.03225806452 + 0.03225806452n + n2 = 0 + 0.03225806452

Combine like terms: -0.03225806452 + 0.03225806452 = 0.00000000000
0.00000000000 + 0.03225806452n + n2 = 0 + 0.03225806452
0.03225806452n + n2 = 0 + 0.03225806452

Combine like terms: 0 + 0.03225806452 = 0.03225806452
0.03225806452n + n2 = 0.03225806452

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

Add '0.25' to each side of the equation.
0.03225806452n + 0.25 + n2 = 0.03225806452 + 0.25

Reorder the terms:
0.25 + 0.03225806452n + n2 = 0.03225806452 + 0.25

Combine like terms: 0.03225806452 + 0.25 = 0.28225806452
0.25 + 0.03225806452n + n2 = 0.28225806452

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.28225806452

Calculate the square root of the right side: 0.531279648

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.531279648 and -0.531279648.

Subproblem 1

n + 0.5 = 0.531279648 Simplifying n + 0.5 = 0.531279648 Reorder the terms: 0.5 + n = 0.531279648 Solving 0.5 + n = 0.531279648 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.531279648 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.531279648 + -0.5 n = 0.531279648 + -0.5 Combine like terms: 0.531279648 + -0.5 = 0.031279648 n = 0.031279648 Simplifying n = 0.031279648

Subproblem 2

n + 0.5 = -0.531279648 Simplifying n + 0.5 = -0.531279648 Reorder the terms: 0.5 + n = -0.531279648 Solving 0.5 + n = -0.531279648 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.531279648 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.531279648 + -0.5 n = -0.531279648 + -0.5 Combine like terms: -0.531279648 + -0.5 = -1.031279648 n = -1.031279648 Simplifying n = -1.031279648

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.031279648, -1.031279648}

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