n^2+2n-27=-6

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


Simplifying
n2 + 2n + -27 = -6

Reorder the terms:
-27 + 2n + n2 = -6

Solving
-27 + 2n + n2 = -6

Solving for variable 'n'.

Reorder the terms:
-27 + 6 + 2n + n2 = -6 + 6

Combine like terms: -27 + 6 = -21
-21 + 2n + n2 = -6 + 6

Combine like terms: -6 + 6 = 0
-21 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 2n + 21 + n2 = 0 + 21

Reorder the terms:
-21 + 21 + 2n + n2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 2n + n2 = 0 + 21
2n + n2 = 0 + 21

Combine like terms: 0 + 21 = 21
2n + n2 = 21

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 21 + 1

Reorder the terms:
1 + 2n + n2 = 21 + 1

Combine like terms: 21 + 1 = 22
1 + 2n + n2 = 22

Factor a perfect square on the left side:
(n + 1)(n + 1) = 22

Calculate the square root of the right side: 4.69041576

Break this problem into two subproblems by setting 
(n + 1) equal to 4.69041576 and -4.69041576.

Subproblem 1

n + 1 = 4.69041576 Simplifying n + 1 = 4.69041576 Reorder the terms: 1 + n = 4.69041576 Solving 1 + n = 4.69041576 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 4.69041576 + -1 Combine like terms: 1 + -1 = 0 0 + n = 4.69041576 + -1 n = 4.69041576 + -1 Combine like terms: 4.69041576 + -1 = 3.69041576 n = 3.69041576 Simplifying n = 3.69041576

Subproblem 2

n + 1 = -4.69041576 Simplifying n + 1 = -4.69041576 Reorder the terms: 1 + n = -4.69041576 Solving 1 + n = -4.69041576 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -4.69041576 + -1 Combine like terms: 1 + -1 = 0 0 + n = -4.69041576 + -1 n = -4.69041576 + -1 Combine like terms: -4.69041576 + -1 = -5.69041576 n = -5.69041576 Simplifying n = -5.69041576

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.69041576, -5.69041576}

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