n^2+6n-9=6

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


Simplifying
n2 + 6n + -9 = 6

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

Solving
-9 + 6n + n2 = 6

Solving for variable 'n'.

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

Combine like terms: -9 + -6 = -15
-15 + 6n + n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-15 + 6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 6n + 15 + n2 = 0 + 15

Reorder the terms:
-15 + 15 + 6n + n2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 6n + n2 = 0 + 15
6n + n2 = 0 + 15

Combine like terms: 0 + 15 = 15
6n + n2 = 15

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 = 15 + 9

Reorder the terms:
9 + 6n + n2 = 15 + 9

Combine like terms: 15 + 9 = 24
9 + 6n + n2 = 24

Factor a perfect square on the left side:
(n + 3)(n + 3) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(n + 3) equal to 4.898979486 and -4.898979486.

Subproblem 1

n + 3 = 4.898979486 Simplifying n + 3 = 4.898979486 Reorder the terms: 3 + n = 4.898979486 Solving 3 + n = 4.898979486 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 = 4.898979486 + -3 Combine like terms: 3 + -3 = 0 0 + n = 4.898979486 + -3 n = 4.898979486 + -3 Combine like terms: 4.898979486 + -3 = 1.898979486 n = 1.898979486 Simplifying n = 1.898979486

Subproblem 2

n + 3 = -4.898979486 Simplifying n + 3 = -4.898979486 Reorder the terms: 3 + n = -4.898979486 Solving 3 + n = -4.898979486 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 = -4.898979486 + -3 Combine like terms: 3 + -3 = 0 0 + n = -4.898979486 + -3 n = -4.898979486 + -3 Combine like terms: -4.898979486 + -3 = -7.898979486 n = -7.898979486 Simplifying n = -7.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.898979486, -7.898979486}

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