n*n+6*n=12

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Solution for n*n+6*n=12 equation:


Simplifying
n * n + 6n = 12

Multiply n * n
n2 + 6n = 12

Reorder the terms:
6n + n2 = 12

Solving
6n + n2 = 12

Solving for variable 'n'.

Reorder the terms:
-12 + 6n + n2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 6n + 12 + n2 = 0 + 12

Reorder the terms:
-12 + 12 + 6n + n2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 6n + n2 = 0 + 12
6n + n2 = 0 + 12

Combine like terms: 0 + 12 = 12
6n + n2 = 12

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 = 12 + 9

Reorder the terms:
9 + 6n + n2 = 12 + 9

Combine like terms: 12 + 9 = 21
9 + 6n + n2 = 21

Factor a perfect square on the left side:
(n + 3)(n + 3) = 21

Calculate the square root of the right side: 4.582575695

Break this problem into two subproblems by setting 
(n + 3) equal to 4.582575695 and -4.582575695.

Subproblem 1

n + 3 = 4.582575695 Simplifying n + 3 = 4.582575695 Reorder the terms: 3 + n = 4.582575695 Solving 3 + n = 4.582575695 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 = 4.582575695 + -3 Combine like terms: 3 + -3 = 0 0 + n = 4.582575695 + -3 n = 4.582575695 + -3 Combine like terms: 4.582575695 + -3 = 1.582575695 n = 1.582575695 Simplifying n = 1.582575695

Subproblem 2

n + 3 = -4.582575695 Simplifying n + 3 = -4.582575695 Reorder the terms: 3 + n = -4.582575695 Solving 3 + n = -4.582575695 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 = -4.582575695 + -3 Combine like terms: 3 + -3 = 0 0 + n = -4.582575695 + -3 n = -4.582575695 + -3 Combine like terms: -4.582575695 + -3 = -7.582575695 n = -7.582575695 Simplifying n = -7.582575695

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.582575695, -7.582575695}

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