n^2-15n-27=0

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Solution for n^2-15n-27=0 equation:


Simplifying
n2 + -15n + -27 = 0

Reorder the terms:
-27 + -15n + n2 = 0

Solving
-27 + -15n + 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 + -15n + 27 + n2 = 0 + 27

Reorder the terms:
-27 + 27 + -15n + n2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -15n + n2 = 0 + 27
-15n + n2 = 0 + 27

Combine like terms: 0 + 27 = 27
-15n + n2 = 27

The n term is -15n.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15n + 56.25 + n2 = 27 + 56.25

Reorder the terms:
56.25 + -15n + n2 = 27 + 56.25

Combine like terms: 27 + 56.25 = 83.25
56.25 + -15n + n2 = 83.25

Factor a perfect square on the left side:
(n + -7.5)(n + -7.5) = 83.25

Calculate the square root of the right side: 9.124143795

Break this problem into two subproblems by setting 
(n + -7.5) equal to 9.124143795 and -9.124143795.

Subproblem 1

n + -7.5 = 9.124143795 Simplifying n + -7.5 = 9.124143795 Reorder the terms: -7.5 + n = 9.124143795 Solving -7.5 + n = 9.124143795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + n = 9.124143795 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + n = 9.124143795 + 7.5 n = 9.124143795 + 7.5 Combine like terms: 9.124143795 + 7.5 = 16.624143795 n = 16.624143795 Simplifying n = 16.624143795

Subproblem 2

n + -7.5 = -9.124143795 Simplifying n + -7.5 = -9.124143795 Reorder the terms: -7.5 + n = -9.124143795 Solving -7.5 + n = -9.124143795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + n = -9.124143795 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + n = -9.124143795 + 7.5 n = -9.124143795 + 7.5 Combine like terms: -9.124143795 + 7.5 = -1.624143795 n = -1.624143795 Simplifying n = -1.624143795

Solution

The solution to the problem is based on the solutions from the subproblems. n = {16.624143795, -1.624143795}

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