6n+n^2=144

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Solution for 6n+n^2=144 equation:


Simplifying
6n + n2 = 144

Solving
6n + n2 = 144

Solving for variable 'n'.

Reorder the terms:
-144 + 6n + n2 = 144 + -144

Combine like terms: 144 + -144 = 0
-144 + 6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '144' to each side of the equation.
-144 + 6n + 144 + n2 = 0 + 144

Reorder the terms:
-144 + 144 + 6n + n2 = 0 + 144

Combine like terms: -144 + 144 = 0
0 + 6n + n2 = 0 + 144
6n + n2 = 0 + 144

Combine like terms: 0 + 144 = 144
6n + n2 = 144

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 = 144 + 9

Reorder the terms:
9 + 6n + n2 = 144 + 9

Combine like terms: 144 + 9 = 153
9 + 6n + n2 = 153

Factor a perfect square on the left side:
(n + 3)(n + 3) = 153

Calculate the square root of the right side: 12.369316877

Break this problem into two subproblems by setting 
(n + 3) equal to 12.369316877 and -12.369316877.

Subproblem 1

n + 3 = 12.369316877 Simplifying n + 3 = 12.369316877 Reorder the terms: 3 + n = 12.369316877 Solving 3 + n = 12.369316877 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 = 12.369316877 + -3 Combine like terms: 3 + -3 = 0 0 + n = 12.369316877 + -3 n = 12.369316877 + -3 Combine like terms: 12.369316877 + -3 = 9.369316877 n = 9.369316877 Simplifying n = 9.369316877

Subproblem 2

n + 3 = -12.369316877 Simplifying n + 3 = -12.369316877 Reorder the terms: 3 + n = -12.369316877 Solving 3 + n = -12.369316877 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 = -12.369316877 + -3 Combine like terms: 3 + -3 = 0 0 + n = -12.369316877 + -3 n = -12.369316877 + -3 Combine like terms: -12.369316877 + -3 = -15.369316877 n = -15.369316877 Simplifying n = -15.369316877

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.369316877, -15.369316877}

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