n^2-4n-78=-9

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Solution for n^2-4n-78=-9 equation:


Simplifying
n2 + -4n + -78 = -9

Reorder the terms:
-78 + -4n + n2 = -9

Solving
-78 + -4n + n2 = -9

Solving for variable 'n'.

Reorder the terms:
-78 + 9 + -4n + n2 = -9 + 9

Combine like terms: -78 + 9 = -69
-69 + -4n + n2 = -9 + 9

Combine like terms: -9 + 9 = 0
-69 + -4n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '69' to each side of the equation.
-69 + -4n + 69 + n2 = 0 + 69

Reorder the terms:
-69 + 69 + -4n + n2 = 0 + 69

Combine like terms: -69 + 69 = 0
0 + -4n + n2 = 0 + 69
-4n + n2 = 0 + 69

Combine like terms: 0 + 69 = 69
-4n + n2 = 69

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 69 + 4

Reorder the terms:
4 + -4n + n2 = 69 + 4

Combine like terms: 69 + 4 = 73
4 + -4n + n2 = 73

Factor a perfect square on the left side:
(n + -2)(n + -2) = 73

Calculate the square root of the right side: 8.544003745

Break this problem into two subproblems by setting 
(n + -2) equal to 8.544003745 and -8.544003745.


Subproblem 1
n + -2 = 8.544003745

Simplifying
n + -2 = 8.544003745

Reorder the terms:
-2 + n = 8.544003745

Solving
-2 + n = 8.544003745

Solving for variable 'n'.

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

Add '2' to each side of the equation.
-2 + 2 + n = 8.544003745 + 2

Combine like terms: -2 + 2 = 0
0 + n = 8.544003745 + 2
n = 8.544003745 + 2

Combine like terms: 8.544003745 + 2 = 10.544003745
n = 10.544003745

Simplifying
n = 10.544003745


Subproblem 2
n + -2 = -8.544003745

Simplifying
n + -2 = -8.544003745

Reorder the terms:
-2 + n = -8.544003745

Solving
-2 + n = -8.544003745

Solving for variable 'n'.

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

Add '2' to each side of the equation.
-2 + 2 + n = -8.544003745 + 2

Combine like terms: -2 + 2 = 0
0 + n = -8.544003745 + 2
n = -8.544003745 + 2

Combine like terms: -8.544003745 + 2 = -6.544003745
n = -6.544003745

Simplifying
n = -6.544003745


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {10.544003745, -6.544003745}

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