n^2-3n-3=9

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Solution for n^2-3n-3=9 equation:


Simplifying
n2 + -3n + -3 = 9

Reorder the terms:
-3 + -3n + n2 = 9

Solving
-3 + -3n + n2 = 9

Solving for variable 'n'.

Reorder the terms:
-3 + -9 + -3n + n2 = 9 + -9

Combine like terms: -3 + -9 = -12
-12 + -3n + n2 = 9 + -9

Combine like terms: 9 + -9 = 0
-12 + -3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + -3n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + -3n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + -3n + n2 = 0 + 12
-3n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
-3n + n2 = 12

The n term is -3n.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3n + 2.25 + n2 = 12 + 2.25

Reorder the terms:
2.25 + -3n + n2 = 12 + 2.25

Combine like terms: 12 + 2.25 = 14.25
2.25 + -3n + n2 = 14.25

Factor a perfect square on the left side:
(n + -1.5)(n + -1.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(n + -1.5) equal to 3.774917218 and -3.774917218.

Subproblem 1

n + -1.5 = 3.774917218 Simplifying n + -1.5 = 3.774917218 Reorder the terms: -1.5 + n = 3.774917218 Solving -1.5 + n = 3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 3.774917218 + 1.5 n = 3.774917218 + 1.5 Combine like terms: 3.774917218 + 1.5 = 5.274917218 n = 5.274917218 Simplifying n = 5.274917218

Subproblem 2

n + -1.5 = -3.774917218 Simplifying n + -1.5 = -3.774917218 Reorder the terms: -1.5 + n = -3.774917218 Solving -1.5 + n = -3.774917218 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -3.774917218 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -3.774917218 + 1.5 n = -3.774917218 + 1.5 Combine like terms: -3.774917218 + 1.5 = -2.274917218 n = -2.274917218 Simplifying n = -2.274917218

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.274917218, -2.274917218}

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