3n^2+n=5

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Solution for 3n^2+n=5 equation:


Simplifying
3n2 + n = 5

Reorder the terms:
n + 3n2 = 5

Solving
n + 3n2 = 5

Solving for variable 'n'.

Reorder the terms:
-5 + n + 3n2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + n + 3n2 = 0

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

Divide each side by '3'.
-1.666666667 + 0.3333333333n + n2 = 0

Move the constant term to the right:

Add '1.666666667' to each side of the equation.
-1.666666667 + 0.3333333333n + 1.666666667 + n2 = 0 + 1.666666667

Reorder the terms:
-1.666666667 + 1.666666667 + 0.3333333333n + n2 = 0 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + 0.3333333333n + n2 = 0 + 1.666666667
0.3333333333n + n2 = 0 + 1.666666667

Combine like terms: 0 + 1.666666667 = 1.666666667
0.3333333333n + n2 = 1.666666667

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.
0.3333333333n + 0.25 + n2 = 1.666666667 + 0.25

Reorder the terms:
0.25 + 0.3333333333n + n2 = 1.666666667 + 0.25

Combine like terms: 1.666666667 + 0.25 = 1.916666667
0.25 + 0.3333333333n + n2 = 1.916666667

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 1.916666667

Calculate the square root of the right side: 1.384437311

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.384437311 and -1.384437311.

Subproblem 1

n + 0.5 = 1.384437311 Simplifying n + 0.5 = 1.384437311 Reorder the terms: 0.5 + n = 1.384437311 Solving 0.5 + n = 1.384437311 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 = 1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.384437311 + -0.5 n = 1.384437311 + -0.5 Combine like terms: 1.384437311 + -0.5 = 0.884437311 n = 0.884437311 Simplifying n = 0.884437311

Subproblem 2

n + 0.5 = -1.384437311 Simplifying n + 0.5 = -1.384437311 Reorder the terms: 0.5 + n = -1.384437311 Solving 0.5 + n = -1.384437311 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 = -1.384437311 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.384437311 + -0.5 n = -1.384437311 + -0.5 Combine like terms: -1.384437311 + -0.5 = -1.884437311 n = -1.884437311 Simplifying n = -1.884437311

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.884437311, -1.884437311}

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