3n-2n(n+2)=-11

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Solution for 3n-2n(n+2)=-11 equation:


Simplifying
3n + -2n(n + 2) = -11

Reorder the terms:
3n + -2n(2 + n) = -11
3n + (2 * -2n + n * -2n) = -11
3n + (-4n + -2n2) = -11

Combine like terms: 3n + -4n = -1n
-1n + -2n2 = -11

Solving
-1n + -2n2 = -11

Solving for variable 'n'.

Reorder the terms:
11 + -1n + -2n2 = -11 + 11

Combine like terms: -11 + 11 = 0
11 + -1n + -2n2 = 0

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

Divide each side by '-2'.
-5.5 + 0.5n + n2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + 0.5n + 5.5 + n2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + 0.5n + n2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 0.5n + n2 = 0 + 5.5
0.5n + n2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
0.5n + n2 = 5.5

The n term is 0.5n.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5n + 0.0625 + n2 = 5.5 + 0.0625

Reorder the terms:
0.0625 + 0.5n + n2 = 5.5 + 0.0625

Combine like terms: 5.5 + 0.0625 = 5.5625
0.0625 + 0.5n + n2 = 5.5625

Factor a perfect square on the left side:
(n + 0.25)(n + 0.25) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(n + 0.25) equal to 2.358495283 and -2.358495283.

Subproblem 1

n + 0.25 = 2.358495283 Simplifying n + 0.25 = 2.358495283 Reorder the terms: 0.25 + n = 2.358495283 Solving 0.25 + n = 2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = 2.358495283 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 2.358495283 + -0.25 n = 2.358495283 + -0.25 Combine like terms: 2.358495283 + -0.25 = 2.108495283 n = 2.108495283 Simplifying n = 2.108495283

Subproblem 2

n + 0.25 = -2.358495283 Simplifying n + 0.25 = -2.358495283 Reorder the terms: 0.25 + n = -2.358495283 Solving 0.25 + n = -2.358495283 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + n = -2.358495283 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -2.358495283 + -0.25 n = -2.358495283 + -0.25 Combine like terms: -2.358495283 + -0.25 = -2.608495283 n = -2.608495283 Simplifying n = -2.608495283

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.108495283, -2.608495283}

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