3n^2-6n-1=0

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Solution for 3n^2-6n-1=0 equation:


Simplifying
3n2 + -6n + -1 = 0

Reorder the terms:
-1 + -6n + 3n2 = 0

Solving
-1 + -6n + 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 + -2n + n2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -2n + 0.3333333333 + n2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -2n + n2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -2n + n2 = 0 + 0.3333333333
-2n + n2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-2n + n2 = 0.3333333333

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 0.3333333333 + 1

Reorder the terms:
1 + -2n + n2 = 0.3333333333 + 1

Combine like terms: 0.3333333333 + 1 = 1.3333333333
1 + -2n + n2 = 1.3333333333

Factor a perfect square on the left side:
(n + -1)(n + -1) = 1.3333333333

Calculate the square root of the right side: 1.154700538

Break this problem into two subproblems by setting 
(n + -1) equal to 1.154700538 and -1.154700538.

Subproblem 1

n + -1 = 1.154700538 Simplifying n + -1 = 1.154700538 Reorder the terms: -1 + n = 1.154700538 Solving -1 + n = 1.154700538 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.154700538 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.154700538 + 1 n = 1.154700538 + 1 Combine like terms: 1.154700538 + 1 = 2.154700538 n = 2.154700538 Simplifying n = 2.154700538

Subproblem 2

n + -1 = -1.154700538 Simplifying n + -1 = -1.154700538 Reorder the terms: -1 + n = -1.154700538 Solving -1 + n = -1.154700538 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.154700538 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.154700538 + 1 n = -1.154700538 + 1 Combine like terms: -1.154700538 + 1 = -0.154700538 n = -0.154700538 Simplifying n = -0.154700538

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.154700538, -0.154700538}

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