5n^2+20n-30=10

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Solution for 5n^2+20n-30=10 equation: 


Simplifying
5n2 + 20n + -30 = 10

Reorder the terms:
-30 + 20n + 5n2 = 10

Solving
-30 + 20n + 5n2 = 10

Solving for variable 'n'.

Reorder the terms:
-30 + -10 + 20n + 5n2 = 10 + -10

Combine like terms: -30 + -10 = -40
-40 + 20n + 5n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-40 + 20n + 5n2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-8 + 4n + n2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-8 + 4n + n2)' equal to zero and attempt to solve:

Simplifying
-8 + 4n + n2 = 0

Solving
-8 + 4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 4n + 8 + n2 = 0 + 8

Reorder the terms:
-8 + 8 + 4n + n2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 4n + n2 = 0 + 8
4n + n2 = 0 + 8

Combine like terms: 0 + 8 = 8
4n + n2 = 8

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 = 8 + 4

Reorder the terms:
4 + 4n + n2 = 8 + 4

Combine like terms: 8 + 4 = 12
4 + 4n + n2 = 12

Factor a perfect square on the left side:
(n + 2)(n + 2) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(n + 2) equal to 3.464101615 and -3.464101615.


Subproblem 1
n + 2 = 3.464101615

Simplifying
n + 2 = 3.464101615

Reorder the terms:
2 + n = 3.464101615

Solving
2 + n = 3.464101615

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 = 3.464101615 + -2

Combine like terms: 2 + -2 = 0
0 + n = 3.464101615 + -2
n = 3.464101615 + -2

Combine like terms: 3.464101615 + -2 = 1.464101615
n = 1.464101615

Simplifying
n = 1.464101615


Subproblem 2
n + 2 = -3.464101615

Simplifying
n + 2 = -3.464101615

Reorder the terms:
2 + n = -3.464101615

Solving
2 + n = -3.464101615

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 = -3.464101615 + -2

Combine like terms: 2 + -2 = 0
0 + n = -3.464101615 + -2
n = -3.464101615 + -2

Combine like terms: -3.464101615 + -2 = -5.464101615
n = -5.464101615

Simplifying
n = -5.464101615


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.464101615, -5.464101615}
Solution
n = {1.464101615, -5.464101615}

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