6n^2=-12n-5

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Solution for 6n^2=-12n-5 equation:


Simplifying
6n2 = -12n + -5

Reorder the terms:
6n2 = -5 + -12n

Solving
6n2 = -5 + -12n

Solving for variable 'n'.

Reorder the terms:
5 + 12n + 6n2 = -5 + -12n + 5 + 12n

Reorder the terms:
5 + 12n + 6n2 = -5 + 5 + -12n + 12n

Combine like terms: -5 + 5 = 0
5 + 12n + 6n2 = 0 + -12n + 12n
5 + 12n + 6n2 = -12n + 12n

Combine like terms: -12n + 12n = 0
5 + 12n + 6n2 = 0

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

Divide each side by '6'.
0.8333333333 + 2n + n2 = 0

Move the constant term to the right:

Add '-0.8333333333' to each side of the equation.
0.8333333333 + 2n + -0.8333333333 + n2 = 0 + -0.8333333333

Reorder the terms:
0.8333333333 + -0.8333333333 + 2n + n2 = 0 + -0.8333333333

Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000
0.0000000000 + 2n + n2 = 0 + -0.8333333333
2n + n2 = 0 + -0.8333333333

Combine like terms: 0 + -0.8333333333 = -0.8333333333
2n + n2 = -0.8333333333

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 = -0.8333333333 + 1

Reorder the terms:
1 + 2n + n2 = -0.8333333333 + 1

Combine like terms: -0.8333333333 + 1 = 0.1666666667
1 + 2n + n2 = 0.1666666667

Factor a perfect square on the left side:
(n + 1)(n + 1) = 0.1666666667

Calculate the square root of the right side: 0.408248291

Break this problem into two subproblems by setting 
(n + 1) equal to 0.408248291 and -0.408248291.

Subproblem 1

n + 1 = 0.408248291 Simplifying n + 1 = 0.408248291 Reorder the terms: 1 + n = 0.408248291 Solving 1 + n = 0.408248291 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 = 0.408248291 + -1 Combine like terms: 1 + -1 = 0 0 + n = 0.408248291 + -1 n = 0.408248291 + -1 Combine like terms: 0.408248291 + -1 = -0.591751709 n = -0.591751709 Simplifying n = -0.591751709

Subproblem 2

n + 1 = -0.408248291 Simplifying n + 1 = -0.408248291 Reorder the terms: 1 + n = -0.408248291 Solving 1 + n = -0.408248291 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 = -0.408248291 + -1 Combine like terms: 1 + -1 = 0 0 + n = -0.408248291 + -1 n = -0.408248291 + -1 Combine like terms: -0.408248291 + -1 = -1.408248291 n = -1.408248291 Simplifying n = -1.408248291

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.591751709, -1.408248291}

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