3n^2+3n-1=0

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Solution for 3n^2+3n-1=0 equation:


Simplifying
3n2 + 3n + -1 = 0

Reorder the terms:
-1 + 3n + 3n2 = 0

Solving
-1 + 3n + 3n2 = 0

Solving for variable 'n'.

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

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

Move the constant term to the right:

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

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

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + n + n2 = 0 + 0.3333333333
n + n2 = 0 + 0.3333333333

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

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.
 + 0.25 + n2 = 0.3333333333 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 0.3333333333 + 0.25

Combine like terms: 0.3333333333 + 0.25 = 0.5833333333
1.25 + n2 = 0.5833333333

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.5833333333

Calculate the square root of the right side: 0.763762616

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.763762616 and -0.763762616.

Subproblem 1

n + 0.5 = 0.763762616 Simplifying n + 0.5 = 0.763762616 Reorder the terms: 0.5 + n = 0.763762616 Solving 0.5 + n = 0.763762616 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 = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.763762616 + -0.5 n = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 n = 0.263762616 Simplifying n = 0.263762616

Subproblem 2

n + 0.5 = -0.763762616 Simplifying n + 0.5 = -0.763762616 Reorder the terms: 0.5 + n = -0.763762616 Solving 0.5 + n = -0.763762616 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 = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.763762616 + -0.5 n = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 n = -1.263762616 Simplifying n = -1.263762616

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.263762616, -1.263762616}

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