4n^2+30n+30=0

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Solution for 4n^2+30n+30=0 equation: 


Simplifying
4n2 + 30n + 30 = 0

Reorder the terms:
30 + 30n + 4n2 = 0

Solving
30 + 30n + 4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(15 + 15n + 2n2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(15 + 15n + 2n2)' equal to zero and attempt to solve:

Simplifying
15 + 15n + 2n2 = 0

Solving
15 + 15n + 2n2 = 0

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

Divide each side by '2'.
7.5 + 7.5n + n2 = 0

Move the constant term to the right:

Add '-7.5' to each side of the equation.
7.5 + 7.5n + -7.5 + n2 = 0 + -7.5

Reorder the terms:
7.5 + -7.5 + 7.5n + n2 = 0 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + 7.5n + n2 = 0 + -7.5
7.5n + n2 = 0 + -7.5

Combine like terms: 0 + -7.5 = -7.5
7.5n + n2 = -7.5

The n term is 7.5n.  Take half its coefficient (3.75).
Square it (14.0625) and add it to both sides.

Add '14.0625' to each side of the equation.
7.5n + 14.0625 + n2 = -7.5 + 14.0625

Reorder the terms:
14.0625 + 7.5n + n2 = -7.5 + 14.0625

Combine like terms: -7.5 + 14.0625 = 6.5625
14.0625 + 7.5n + n2 = 6.5625

Factor a perfect square on the left side:
(n + 3.75)(n + 3.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(n + 3.75) equal to 2.561737691 and -2.561737691.


Subproblem 1
n + 3.75 = 2.561737691

Simplifying
n + 3.75 = 2.561737691

Reorder the terms:
3.75 + n = 2.561737691

Solving
3.75 + n = 2.561737691

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-3.75' to each side of the equation.
3.75 + -3.75 + n = 2.561737691 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + n = 2.561737691 + -3.75
n = 2.561737691 + -3.75

Combine like terms: 2.561737691 + -3.75 = -1.188262309
n = -1.188262309

Simplifying
n = -1.188262309


Subproblem 2
n + 3.75 = -2.561737691

Simplifying
n + 3.75 = -2.561737691

Reorder the terms:
3.75 + n = -2.561737691

Solving
3.75 + n = -2.561737691

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-3.75' to each side of the equation.
3.75 + -3.75 + n = -2.561737691 + -3.75

Combine like terms: 3.75 + -3.75 = 0.00
0.00 + n = -2.561737691 + -3.75
n = -2.561737691 + -3.75

Combine like terms: -2.561737691 + -3.75 = -6.311737691
n = -6.311737691

Simplifying
n = -6.311737691


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-1.188262309, -6.311737691}
Solution
n = {-1.188262309, -6.311737691}

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