n^2-3n-27=0

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Solution for n^2-3n-27=0 equation:


Simplifying
n2 + -3n + -27 = 0

Reorder the terms:
-27 + -3n + n2 = 0

Solving
-27 + -3n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '27' to each side of the equation.
-27 + -3n + 27 + n2 = 0 + 27

Reorder the terms:
-27 + 27 + -3n + n2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -3n + n2 = 0 + 27
-3n + n2 = 0 + 27

Combine like terms: 0 + 27 = 27
-3n + n2 = 27

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 27 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 27 + 2.25

Combine like terms: 27 + 2.25 = 29.25
2.25 + -3n + n2 = 29.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 29.25

Calculate the square root of the right side: 5.408326913

Break this problem into two subproblems by setting 
(n + -1.5) equal to 5.408326913 and -5.408326913.

Subproblem 1

n + -1.5 = 5.408326913 Simplifying n + -1.5 = 5.408326913 Reorder the terms: -1.5 + n = 5.408326913 Solving -1.5 + n = 5.408326913 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 5.408326913 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 5.408326913 + 1.5 n = 5.408326913 + 1.5 Combine like terms: 5.408326913 + 1.5 = 6.908326913 n = 6.908326913 Simplifying n = 6.908326913

Subproblem 2

n + -1.5 = -5.408326913 Simplifying n + -1.5 = -5.408326913 Reorder the terms: -1.5 + n = -5.408326913 Solving -1.5 + n = -5.408326913 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -5.408326913 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -5.408326913 + 1.5 n = -5.408326913 + 1.5 Combine like terms: -5.408326913 + 1.5 = -3.908326913 n = -3.908326913 Simplifying n = -3.908326913

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.908326913, -3.908326913}

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