30n(10n+6)=12n+120

Solution for 30n(10n+6)=12n+120 equation:

Simplifying
30n(10n + 6) = 12n + 120

Reorder the terms:
30n(6 + 10n) = 12n + 120
(6 * 30n + 10n * 30n) = 12n + 120
(180n + 300n2) = 12n + 120

Reorder the terms:
180n + 300n2 = 120 + 12n

Solving
180n + 300n2 = 120 + 12n

Solving for variable 'n'.

Reorder the terms:
-120 + 180n + -12n + 300n2 = 120 + 12n + -120 + -12n

Combine like terms: 180n + -12n = 168n
-120 + 168n + 300n2 = 120 + 12n + -120 + -12n

Reorder the terms:
-120 + 168n + 300n2 = 120 + -120 + 12n + -12n

Combine like terms: 120 + -120 = 0
-120 + 168n + 300n2 = 0 + 12n + -12n
-120 + 168n + 300n2 = 12n + -12n

Combine like terms: 12n + -12n = 0
-120 + 168n + 300n2 = 0

Factor out the Greatest Common Factor (GCF), '12'.
12(-10 + 14n + 25n2) = 0

Ignore the factor 12.
Subproblem 1
Set the factor '(-10 + 14n + 25n2)' equal to zero and attempt to solve:

Simplifying
-10 + 14n + 25n2 = 0

Solving
-10 + 14n + 25n2 = 0

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

Divide each side by '25'.
-0.4 + 0.56n + n2 = 0

Move the constant term to the right:

Add '0.4' to each side of the equation.
-0.4 + 0.56n + 0.4 + n2 = 0 + 0.4

Reorder the terms:
-0.4 + 0.4 + 0.56n + n2 = 0 + 0.4

Combine like terms: -0.4 + 0.4 = 0.0
0.0 + 0.56n + n2 = 0 + 0.4
0.56n + n2 = 0 + 0.4

Combine like terms: 0 + 0.4 = 0.4
0.56n + n2 = 0.4

The n term is 0.56n.  Take half its coefficient (0.28).
Square it (0.0784) and add it to both sides.

Add '0.0784' to each side of the equation.
0.56n + 0.0784 + n2 = 0.4 + 0.0784

Reorder the terms:
0.0784 + 0.56n + n2 = 0.4 + 0.0784

Combine like terms: 0.4 + 0.0784 = 0.4784
0.0784 + 0.56n + n2 = 0.4784

Factor a perfect square on the left side:
(n + 0.28)(n + 0.28) = 0.4784

Calculate the square root of the right side: 0.691664659

Break this problem into two subproblems by setting 
(n + 0.28) equal to 0.691664659 and -0.691664659.

Subproblem 1
n + 0.28 = 0.691664659

Simplifying
n + 0.28 = 0.691664659

Reorder the terms:
0.28 + n = 0.691664659

Solving
0.28 + n = 0.691664659

Solving for variable 'n'.

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

Add '-0.28' to each side of the equation.
0.28 + -0.28 + n = 0.691664659 + -0.28

Combine like terms: 0.28 + -0.28 = 0.00
0.00 + n = 0.691664659 + -0.28
n = 0.691664659 + -0.28

Combine like terms: 0.691664659 + -0.28 = 0.411664659
n = 0.411664659

Simplifying
n = 0.411664659

Subproblem 2
n + 0.28 = -0.691664659

Simplifying
n + 0.28 = -0.691664659

Reorder the terms:
0.28 + n = -0.691664659

Solving
0.28 + n = -0.691664659

Solving for variable 'n'.

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

Add '-0.28' to each side of the equation.
0.28 + -0.28 + n = -0.691664659 + -0.28

Combine like terms: 0.28 + -0.28 = 0.00
0.00 + n = -0.691664659 + -0.28
n = -0.691664659 + -0.28

Combine like terms: -0.691664659 + -0.28 = -0.971664659
n = -0.971664659

Simplifying
n = -0.971664659

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.411664659, -0.971664659}
Solution
n = {0.411664659, -0.971664659}