3n^2+12n=-5

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Solution for 3n^2+12n=-5 equation:


Simplifying
3n2 + 12n = -5

Reorder the terms:
12n + 3n2 = -5

Solving
12n + 3n2 = -5

Solving for variable 'n'.

Reorder the terms:
5 + 12n + 3n2 = -5 + 5

Combine like terms: -5 + 5 = 0
5 + 12n + 3n2 = 0

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

Divide each side by '3'.
1.666666667 + 4n + n2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + 4n + -1.666666667 + n2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + 4n + n2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + 4n + n2 = 0 + -1.666666667
4n + n2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
4n + n2 = -1.666666667

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 = -1.666666667 + 4

Reorder the terms:
4 + 4n + n2 = -1.666666667 + 4

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

Factor a perfect square on the left side:
(n + 2)(n + 2) = 2.333333333

Calculate the square root of the right side: 1.527525232

Break this problem into two subproblems by setting 
(n + 2) equal to 1.527525232 and -1.527525232.

Subproblem 1

n + 2 = 1.527525232 Simplifying n + 2 = 1.527525232 Reorder the terms: 2 + n = 1.527525232 Solving 2 + n = 1.527525232 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 = 1.527525232 + -2 Combine like terms: 2 + -2 = 0 0 + n = 1.527525232 + -2 n = 1.527525232 + -2 Combine like terms: 1.527525232 + -2 = -0.472474768 n = -0.472474768 Simplifying n = -0.472474768

Subproblem 2

n + 2 = -1.527525232 Simplifying n + 2 = -1.527525232 Reorder the terms: 2 + n = -1.527525232 Solving 2 + n = -1.527525232 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 = -1.527525232 + -2 Combine like terms: 2 + -2 = 0 0 + n = -1.527525232 + -2 n = -1.527525232 + -2 Combine like terms: -1.527525232 + -2 = -3.527525232 n = -3.527525232 Simplifying n = -3.527525232

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.472474768, -3.527525232}

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