3n^2+12n+9=-1

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Solution for 3n^2+12n+9=-1 equation:


Simplifying
3n2 + 12n + 9 = -1

Reorder the terms:
9 + 12n + 3n2 = -1

Solving
9 + 12n + 3n2 = -1

Solving for variable 'n'.

Reorder the terms:
9 + 1 + 12n + 3n2 = -1 + 1

Combine like terms: 9 + 1 = 10
10 + 12n + 3n2 = -1 + 1

Combine like terms: -1 + 1 = 0
10 + 12n + 3n2 = 0

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

Divide each side by '3'.
3.333333333 + 4n + n2 = 0

Move the constant term to the right:

Add '-3.333333333' to each side of the equation.
3.333333333 + 4n + -3.333333333 + n2 = 0 + -3.333333333

Reorder the terms:
3.333333333 + -3.333333333 + 4n + n2 = 0 + -3.333333333

Combine like terms: 3.333333333 + -3.333333333 = 0.000000000
0.000000000 + 4n + n2 = 0 + -3.333333333
4n + n2 = 0 + -3.333333333

Combine like terms: 0 + -3.333333333 = -3.333333333
4n + n2 = -3.333333333

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = -3.333333333 + 4

Reorder the terms:
4 + 4n + n2 = -3.333333333 + 4

Combine like terms: -3.333333333 + 4 = 0.666666667
4 + 4n + n2 = 0.666666667

Factor a perfect square on the left side:
(n + 2)(n + 2) = 0.666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(n + 2) equal to 0.816496581 and -0.816496581.

Subproblem 1

n + 2 = 0.816496581 Simplifying n + 2 = 0.816496581 Reorder the terms: 2 + n = 0.816496581 Solving 2 + n = 0.816496581 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 0.816496581 + -2 Combine like terms: 2 + -2 = 0 0 + n = 0.816496581 + -2 n = 0.816496581 + -2 Combine like terms: 0.816496581 + -2 = -1.183503419 n = -1.183503419 Simplifying n = -1.183503419

Subproblem 2

n + 2 = -0.816496581 Simplifying n + 2 = -0.816496581 Reorder the terms: 2 + n = -0.816496581 Solving 2 + n = -0.816496581 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -0.816496581 + -2 Combine like terms: 2 + -2 = 0 0 + n = -0.816496581 + -2 n = -0.816496581 + -2 Combine like terms: -0.816496581 + -2 = -2.816496581 n = -2.816496581 Simplifying n = -2.816496581

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.183503419, -2.816496581}

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