n^2-6n-9=11

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Solution for n^2-6n-9=11 equation:


Simplifying
n2 + -6n + -9 = 11

Reorder the terms:
-9 + -6n + n2 = 11

Solving
-9 + -6n + n2 = 11

Solving for variable 'n'.

Reorder the terms:
-9 + -11 + -6n + n2 = 11 + -11

Combine like terms: -9 + -11 = -20
-20 + -6n + n2 = 11 + -11

Combine like terms: 11 + -11 = 0
-20 + -6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -6n + 20 + n2 = 0 + 20

Reorder the terms:
-20 + 20 + -6n + n2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -6n + n2 = 0 + 20
-6n + n2 = 0 + 20

Combine like terms: 0 + 20 = 20
-6n + n2 = 20

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 = 20 + 9

Reorder the terms:
9 + -6n + n2 = 20 + 9

Combine like terms: 20 + 9 = 29
9 + -6n + n2 = 29

Factor a perfect square on the left side:
(n + -3)(n + -3) = 29

Calculate the square root of the right side: 5.385164807

Break this problem into two subproblems by setting 
(n + -3) equal to 5.385164807 and -5.385164807.

Subproblem 1

n + -3 = 5.385164807 Simplifying n + -3 = 5.385164807 Reorder the terms: -3 + n = 5.385164807 Solving -3 + n = 5.385164807 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 = 5.385164807 + 3 Combine like terms: -3 + 3 = 0 0 + n = 5.385164807 + 3 n = 5.385164807 + 3 Combine like terms: 5.385164807 + 3 = 8.385164807 n = 8.385164807 Simplifying n = 8.385164807

Subproblem 2

n + -3 = -5.385164807 Simplifying n + -3 = -5.385164807 Reorder the terms: -3 + n = -5.385164807 Solving -3 + n = -5.385164807 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 = -5.385164807 + 3 Combine like terms: -3 + 3 = 0 0 + n = -5.385164807 + 3 n = -5.385164807 + 3 Combine like terms: -5.385164807 + 3 = -2.385164807 n = -2.385164807 Simplifying n = -2.385164807

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.385164807, -2.385164807}

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