n^2-6n-77=-6

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Solution for n^2-6n-77=-6 equation:


Simplifying
n2 + -6n + -77 = -6

Reorder the terms:
-77 + -6n + n2 = -6

Solving
-77 + -6n + n2 = -6

Solving for variable 'n'.

Reorder the terms:
-77 + 6 + -6n + n2 = -6 + 6

Combine like terms: -77 + 6 = -71
-71 + -6n + n2 = -6 + 6

Combine like terms: -6 + 6 = 0
-71 + -6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '71' to each side of the equation.
-71 + -6n + 71 + n2 = 0 + 71

Reorder the terms:
-71 + 71 + -6n + n2 = 0 + 71

Combine like terms: -71 + 71 = 0
0 + -6n + n2 = 0 + 71
-6n + n2 = 0 + 71

Combine like terms: 0 + 71 = 71
-6n + n2 = 71

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = 71 + 9

Reorder the terms:
9 + -6n + n2 = 71 + 9

Combine like terms: 71 + 9 = 80
9 + -6n + n2 = 80

Factor a perfect square on the left side:
(n + -3)(n + -3) = 80

Calculate the square root of the right side: 8.94427191

Break this problem into two subproblems by setting 
(n + -3) equal to 8.94427191 and -8.94427191.

Subproblem 1

n + -3 = 8.94427191 Simplifying n + -3 = 8.94427191 Reorder the terms: -3 + n = 8.94427191 Solving -3 + n = 8.94427191 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 8.94427191 + 3 Combine like terms: -3 + 3 = 0 0 + n = 8.94427191 + 3 n = 8.94427191 + 3 Combine like terms: 8.94427191 + 3 = 11.94427191 n = 11.94427191 Simplifying n = 11.94427191

Subproblem 2

n + -3 = -8.94427191 Simplifying n + -3 = -8.94427191 Reorder the terms: -3 + n = -8.94427191 Solving -3 + n = -8.94427191 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -8.94427191 + 3 Combine like terms: -3 + 3 = 0 0 + n = -8.94427191 + 3 n = -8.94427191 + 3 Combine like terms: -8.94427191 + 3 = -5.94427191 n = -5.94427191 Simplifying n = -5.94427191

Solution

The solution to the problem is based on the solutions from the subproblems. n = {11.94427191, -5.94427191}

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