4(n+2)(n+2)=280

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Solution for 4(n+2)(n+2)=280 equation:


Simplifying
4(n + 2)(n + 2) = 280

Reorder the terms:
4(2 + n)(n + 2) = 280

Reorder the terms:
4(2 + n)(2 + n) = 280

Multiply (2 + n) * (2 + n)
4(2(2 + n) + n(2 + n)) = 280
4((2 * 2 + n * 2) + n(2 + n)) = 280
4((4 + 2n) + n(2 + n)) = 280
4(4 + 2n + (2 * n + n * n)) = 280
4(4 + 2n + (2n + n2)) = 280

Combine like terms: 2n + 2n = 4n
4(4 + 4n + n2) = 280
(4 * 4 + 4n * 4 + n2 * 4) = 280
(16 + 16n + 4n2) = 280

Solving
16 + 16n + 4n2 = 280

Solving for variable 'n'.

Reorder the terms:
16 + -280 + 16n + 4n2 = 280 + -280

Combine like terms: 16 + -280 = -264
-264 + 16n + 4n2 = 280 + -280

Combine like terms: 280 + -280 = 0
-264 + 16n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-66 + 4n + n2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-66 + 4n + n2)' equal to zero and attempt to solve:

Simplifying
-66 + 4n + n2 = 0

Solving
-66 + 4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '66' to each side of the equation.
-66 + 4n + 66 + n2 = 0 + 66

Reorder the terms:
-66 + 66 + 4n + n2 = 0 + 66

Combine like terms: -66 + 66 = 0
0 + 4n + n2 = 0 + 66
4n + n2 = 0 + 66

Combine like terms: 0 + 66 = 66
4n + n2 = 66

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 = 66 + 4

Reorder the terms:
4 + 4n + n2 = 66 + 4

Combine like terms: 66 + 4 = 70
4 + 4n + n2 = 70

Factor a perfect square on the left side:
(n + 2)(n + 2) = 70

Calculate the square root of the right side: 8.366600265

Break this problem into two subproblems by setting 
(n + 2) equal to 8.366600265 and -8.366600265.


Subproblem 1
n + 2 = 8.366600265

Simplifying
n + 2 = 8.366600265

Reorder the terms:
2 + n = 8.366600265

Solving
2 + n = 8.366600265

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 = 8.366600265 + -2

Combine like terms: 2 + -2 = 0
0 + n = 8.366600265 + -2
n = 8.366600265 + -2

Combine like terms: 8.366600265 + -2 = 6.366600265
n = 6.366600265

Simplifying
n = 6.366600265


Subproblem 2
n + 2 = -8.366600265

Simplifying
n + 2 = -8.366600265

Reorder the terms:
2 + n = -8.366600265

Solving
2 + n = -8.366600265

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 = -8.366600265 + -2

Combine like terms: 2 + -2 = 0
0 + n = -8.366600265 + -2
n = -8.366600265 + -2

Combine like terms: -8.366600265 + -2 = -10.366600265
n = -10.366600265

Simplifying
n = -10.366600265


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.366600265, -10.366600265}
Solution
n = {6.366600265, -10.366600265}

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