n^2+6n+27=47

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Solution for n^2+6n+27=47 equation:


Simplifying
n2 + 6n + 27 = 47

Reorder the terms:
27 + 6n + n2 = 47

Solving
27 + 6n + n2 = 47

Solving for variable 'n'.

Reorder the terms:
27 + -47 + 6n + n2 = 47 + -47

Combine like terms: 27 + -47 = -20
-20 + 6n + n2 = 47 + -47

Combine like terms: 47 + -47 = 0
-20 + 6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 6n + 20 + n2 = 0 + 20

Reorder the terms:
-20 + 20 + 6n + n2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 6n + n2 = 0 + 20
6n + n2 = 0 + 20

Combine like terms: 0 + 20 = 20
6n + n2 = 20

The n term is 6n.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6n + 9 + n2 = 20 + 9

Reorder the terms:
9 + 6n + n2 = 20 + 9

Combine like terms: 20 + 9 = 29
9 + 6n + n2 = 29

Factor a perfect square on the left side:
(n + 3)(n + 3) = 29

Calculate the square root of the right side: 5.385164807

Break this problem into two subproblems by setting 
(n + 3) equal to 5.385164807 and -5.385164807.

Subproblem 1

n + 3 = 5.385164807 Simplifying n + 3 = 5.385164807 Reorder the terms: 3 + n = 5.385164807 Solving 3 + n = 5.385164807 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = 5.385164807 + -3 Combine like terms: 3 + -3 = 0 0 + n = 5.385164807 + -3 n = 5.385164807 + -3 Combine like terms: 5.385164807 + -3 = 2.385164807 n = 2.385164807 Simplifying n = 2.385164807

Subproblem 2

n + 3 = -5.385164807 Simplifying n + 3 = -5.385164807 Reorder the terms: 3 + n = -5.385164807 Solving 3 + n = -5.385164807 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + n = -5.385164807 + -3 Combine like terms: 3 + -3 = 0 0 + n = -5.385164807 + -3 n = -5.385164807 + -3 Combine like terms: -5.385164807 + -3 = -8.385164807 n = -8.385164807 Simplifying n = -8.385164807

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.385164807, -8.385164807}

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