3n^2+9n+4=0

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Solution for 3n^2+9n+4=0 equation:


Simplifying
3n2 + 9n + 4 = 0

Reorder the terms:
4 + 9n + 3n2 = 0

Solving
4 + 9n + 3n2 = 0

Solving for variable 'n'.

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

Divide each side by '3'.
1.333333333 + 3n + n2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + 3n + -1.333333333 + n2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + 3n + n2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + 3n + n2 = 0 + -1.333333333
3n + n2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
3n + n2 = -1.333333333

The n term is 3n.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3n + 2.25 + n2 = -1.333333333 + 2.25

Reorder the terms:
2.25 + 3n + n2 = -1.333333333 + 2.25

Combine like terms: -1.333333333 + 2.25 = 0.916666667
2.25 + 3n + n2 = 0.916666667

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 0.916666667

Calculate the square root of the right side: 0.957427108

Break this problem into two subproblems by setting 
(n + 1.5) equal to 0.957427108 and -0.957427108.

Subproblem 1

n + 1.5 = 0.957427108 Simplifying n + 1.5 = 0.957427108 Reorder the terms: 1.5 + n = 0.957427108 Solving 1.5 + n = 0.957427108 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = 0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 0.957427108 + -1.5 n = 0.957427108 + -1.5 Combine like terms: 0.957427108 + -1.5 = -0.542572892 n = -0.542572892 Simplifying n = -0.542572892

Subproblem 2

n + 1.5 = -0.957427108 Simplifying n + 1.5 = -0.957427108 Reorder the terms: 1.5 + n = -0.957427108 Solving 1.5 + n = -0.957427108 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + n = -0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -0.957427108 + -1.5 n = -0.957427108 + -1.5 Combine like terms: -0.957427108 + -1.5 = -2.457427108 n = -2.457427108 Simplifying n = -2.457427108

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.542572892, -2.457427108}

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