2n^2+12n=-9

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


Simplifying
2n2 + 12n = -9

Reorder the terms:
12n + 2n2 = -9

Solving
12n + 2n2 = -9

Solving for variable 'n'.

Reorder the terms:
9 + 12n + 2n2 = -9 + 9

Combine like terms: -9 + 9 = 0
9 + 12n + 2n2 = 0

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

Divide each side by '2'.
4.5 + 6n + n2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 6n + -4.5 + n2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 6n + n2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 6n + n2 = 0 + -4.5
6n + n2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
6n + n2 = -4.5

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = -4.5 + 9

Reorder the terms:
9 + 6n + n2 = -4.5 + 9

Combine like terms: -4.5 + 9 = 4.5
9 + 6n + n2 = 4.5

Factor a perfect square on the left side:
(n + 3)(n + 3) = 4.5

Calculate the square root of the right side: 2.121320344

Break this problem into two subproblems by setting 
(n + 3) equal to 2.121320344 and -2.121320344.

Subproblem 1

n + 3 = 2.121320344 Simplifying n + 3 = 2.121320344 Reorder the terms: 3 + n = 2.121320344 Solving 3 + n = 2.121320344 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 2.121320344 + -3 Combine like terms: 3 + -3 = 0 0 + n = 2.121320344 + -3 n = 2.121320344 + -3 Combine like terms: 2.121320344 + -3 = -0.878679656 n = -0.878679656 Simplifying n = -0.878679656

Subproblem 2

n + 3 = -2.121320344 Simplifying n + 3 = -2.121320344 Reorder the terms: 3 + n = -2.121320344 Solving 3 + n = -2.121320344 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -2.121320344 + -3 Combine like terms: 3 + -3 = 0 0 + n = -2.121320344 + -3 n = -2.121320344 + -3 Combine like terms: -2.121320344 + -3 = -5.121320344 n = -5.121320344 Simplifying n = -5.121320344

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.878679656, -5.121320344}

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