2n(n+3)=23

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Solution for 2n(n+3)=23 equation:


Simplifying
2n(n + 3) = 23

Reorder the terms:
2n(3 + n) = 23
(3 * 2n + n * 2n) = 23
(6n + 2n2) = 23

Solving
6n + 2n2 = 23

Solving for variable 'n'.

Reorder the terms:
-23 + 6n + 2n2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + 6n + 2n2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-11.5 + 3n + n2 = 0

Move the constant term to the right:

Add '11.5' to each side of the equation.
-11.5 + 3n + 11.5 + n2 = 0 + 11.5

Reorder the terms:
-11.5 + 11.5 + 3n + n2 = 0 + 11.5

Combine like terms: -11.5 + 11.5 = 0.0
0.0 + 3n + n2 = 0 + 11.5
3n + n2 = 0 + 11.5

Combine like terms: 0 + 11.5 = 11.5
3n + n2 = 11.5

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 = 11.5 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 11.5 + 2.25

Combine like terms: 11.5 + 2.25 = 13.75
2.25 + 3n + n2 = 13.75

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 13.75

Calculate the square root of the right side: 3.708099244

Break this problem into two subproblems by setting 
(n + 1.5) equal to 3.708099244 and -3.708099244.

Subproblem 1

n + 1.5 = 3.708099244 Simplifying n + 1.5 = 3.708099244 Reorder the terms: 1.5 + n = 3.708099244 Solving 1.5 + n = 3.708099244 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 = 3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 3.708099244 + -1.5 n = 3.708099244 + -1.5 Combine like terms: 3.708099244 + -1.5 = 2.208099244 n = 2.208099244 Simplifying n = 2.208099244

Subproblem 2

n + 1.5 = -3.708099244 Simplifying n + 1.5 = -3.708099244 Reorder the terms: 1.5 + n = -3.708099244 Solving 1.5 + n = -3.708099244 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 = -3.708099244 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -3.708099244 + -1.5 n = -3.708099244 + -1.5 Combine like terms: -3.708099244 + -1.5 = -5.208099244 n = -5.208099244 Simplifying n = -5.208099244

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.208099244, -5.208099244}

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