n^2-5n-12=0

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Solution for n^2-5n-12=0 equation: 


Simplifying
n2 + -5n + -12 = 0

Reorder the terms:
-12 + -5n + n2 = 0

Solving
-12 + -5n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + -5n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + -5n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + -5n + n2 = 0 + 12
-5n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
-5n + n2 = 12

The n term is -5n.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5n + 6.25 + n2 = 12 + 6.25

Reorder the terms:
6.25 + -5n + n2 = 12 + 6.25

Combine like terms: 12 + 6.25 = 18.25
6.25 + -5n + n2 = 18.25

Factor a perfect square on the left side:
(n + -2.5)(n + -2.5) = 18.25

Calculate the square root of the right side: 4.272001873

Break this problem into two subproblems by setting 
(n + -2.5) equal to 4.272001873 and -4.272001873.


Subproblem 1
n + -2.5 = 4.272001873

Simplifying
n + -2.5 = 4.272001873

Reorder the terms:
-2.5 + n = 4.272001873

Solving
-2.5 + n = 4.272001873

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = 4.272001873 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = 4.272001873 + 2.5
n = 4.272001873 + 2.5

Combine like terms: 4.272001873 + 2.5 = 6.772001873
n = 6.772001873

Simplifying
n = 6.772001873


Subproblem 2
n + -2.5 = -4.272001873

Simplifying
n + -2.5 = -4.272001873

Reorder the terms:
-2.5 + n = -4.272001873

Solving
-2.5 + n = -4.272001873

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '2.5' to each side of the equation.
-2.5 + 2.5 + n = -4.272001873 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + n = -4.272001873 + 2.5
n = -4.272001873 + 2.5

Combine like terms: -4.272001873 + 2.5 = -1.772001873
n = -1.772001873

Simplifying
n = -1.772001873


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.772001873, -1.772001873}

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