3n^2=-12n-7

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Solution for 3n^2=-12n-7 equation:


Simplifying
3n2 = -12n + -7

Reorder the terms:
3n2 = -7 + -12n

Solving
3n2 = -7 + -12n

Solving for variable 'n'.

Reorder the terms:
7 + 12n + 3n2 = -7 + -12n + 7 + 12n

Reorder the terms:
7 + 12n + 3n2 = -7 + 7 + -12n + 12n

Combine like terms: -7 + 7 = 0
7 + 12n + 3n2 = 0 + -12n + 12n
7 + 12n + 3n2 = -12n + 12n

Combine like terms: -12n + 12n = 0
7 + 12n + 3n2 = 0

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

Divide each side by '3'.
2.333333333 + 4n + n2 = 0

Move the constant term to the right:

Add '-2.333333333' to each side of the equation.
2.333333333 + 4n + -2.333333333 + n2 = 0 + -2.333333333

Reorder the terms:
2.333333333 + -2.333333333 + 4n + n2 = 0 + -2.333333333

Combine like terms: 2.333333333 + -2.333333333 = 0.000000000
0.000000000 + 4n + n2 = 0 + -2.333333333
4n + n2 = 0 + -2.333333333

Combine like terms: 0 + -2.333333333 = -2.333333333
4n + n2 = -2.333333333

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 = -2.333333333 + 4

Reorder the terms:
4 + 4n + n2 = -2.333333333 + 4

Combine like terms: -2.333333333 + 4 = 1.666666667
4 + 4n + n2 = 1.666666667

Factor a perfect square on the left side:
(n + 2)(n + 2) = 1.666666667

Calculate the square root of the right side: 1.290994449

Break this problem into two subproblems by setting 
(n + 2) equal to 1.290994449 and -1.290994449.

Subproblem 1

n + 2 = 1.290994449 Simplifying n + 2 = 1.290994449 Reorder the terms: 2 + n = 1.290994449 Solving 2 + n = 1.290994449 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 = 1.290994449 + -2 Combine like terms: 2 + -2 = 0 0 + n = 1.290994449 + -2 n = 1.290994449 + -2 Combine like terms: 1.290994449 + -2 = -0.709005551 n = -0.709005551 Simplifying n = -0.709005551

Subproblem 2

n + 2 = -1.290994449 Simplifying n + 2 = -1.290994449 Reorder the terms: 2 + n = -1.290994449 Solving 2 + n = -1.290994449 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 = -1.290994449 + -2 Combine like terms: 2 + -2 = 0 0 + n = -1.290994449 + -2 n = -1.290994449 + -2 Combine like terms: -1.290994449 + -2 = -3.290994449 n = -3.290994449 Simplifying n = -3.290994449

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.709005551, -3.290994449}

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