n^2-18n-33=6

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Solution for n^2-18n-33=6 equation:


Simplifying
n2 + -18n + -33 = 6

Reorder the terms:
-33 + -18n + n2 = 6

Solving
-33 + -18n + n2 = 6

Solving for variable 'n'.

Reorder the terms:
-33 + -6 + -18n + n2 = 6 + -6

Combine like terms: -33 + -6 = -39
-39 + -18n + n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-39 + -18n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + -18n + 39 + n2 = 0 + 39

Reorder the terms:
-39 + 39 + -18n + n2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + -18n + n2 = 0 + 39
-18n + n2 = 0 + 39

Combine like terms: 0 + 39 = 39
-18n + n2 = 39

The n term is -18n.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18n + 81 + n2 = 39 + 81

Reorder the terms:
81 + -18n + n2 = 39 + 81

Combine like terms: 39 + 81 = 120
81 + -18n + n2 = 120

Factor a perfect square on the left side:
(n + -9)(n + -9) = 120

Calculate the square root of the right side: 10.95445115

Break this problem into two subproblems by setting 
(n + -9) equal to 10.95445115 and -10.95445115.

Subproblem 1

n + -9 = 10.95445115 Simplifying n + -9 = 10.95445115 Reorder the terms: -9 + n = 10.95445115 Solving -9 + n = 10.95445115 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = 10.95445115 + 9 Combine like terms: -9 + 9 = 0 0 + n = 10.95445115 + 9 n = 10.95445115 + 9 Combine like terms: 10.95445115 + 9 = 19.95445115 n = 19.95445115 Simplifying n = 19.95445115

Subproblem 2

n + -9 = -10.95445115 Simplifying n + -9 = -10.95445115 Reorder the terms: -9 + n = -10.95445115 Solving -9 + n = -10.95445115 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + n = -10.95445115 + 9 Combine like terms: -9 + 9 = 0 0 + n = -10.95445115 + 9 n = -10.95445115 + 9 Combine like terms: -10.95445115 + 9 = -1.95445115 n = -1.95445115 Simplifying n = -1.95445115

Solution

The solution to the problem is based on the solutions from the subproblems. n = {19.95445115, -1.95445115}

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