n(n+2)+(n+4)=42

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Solution for n(n+2)+(n+4)=42 equation:


Simplifying
n(n + 2) + (n + 4) = 42

Reorder the terms:
n(2 + n) + (n + 4) = 42
(2 * n + n * n) + (n + 4) = 42
(2n + n2) + (n + 4) = 42

Reorder the terms:
2n + n2 + (4 + n) = 42

Remove parenthesis around (4 + n)
2n + n2 + 4 + n = 42

Reorder the terms:
4 + 2n + n + n2 = 42

Combine like terms: 2n + n = 3n
4 + 3n + n2 = 42

Solving
4 + 3n + n2 = 42

Solving for variable 'n'.

Reorder the terms:
4 + -42 + 3n + n2 = 42 + -42

Combine like terms: 4 + -42 = -38
-38 + 3n + n2 = 42 + -42

Combine like terms: 42 + -42 = 0
-38 + 3n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '38' to each side of the equation.
-38 + 3n + 38 + n2 = 0 + 38

Reorder the terms:
-38 + 38 + 3n + n2 = 0 + 38

Combine like terms: -38 + 38 = 0
0 + 3n + n2 = 0 + 38
3n + n2 = 0 + 38

Combine like terms: 0 + 38 = 38
3n + n2 = 38

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 = 38 + 2.25

Reorder the terms:
2.25 + 3n + n2 = 38 + 2.25

Combine like terms: 38 + 2.25 = 40.25
2.25 + 3n + n2 = 40.25

Factor a perfect square on the left side:
(n + 1.5)(n + 1.5) = 40.25

Calculate the square root of the right side: 6.34428877

Break this problem into two subproblems by setting 
(n + 1.5) equal to 6.34428877 and -6.34428877.

Subproblem 1

n + 1.5 = 6.34428877 Simplifying n + 1.5 = 6.34428877 Reorder the terms: 1.5 + n = 6.34428877 Solving 1.5 + n = 6.34428877 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 = 6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 6.34428877 + -1.5 n = 6.34428877 + -1.5 Combine like terms: 6.34428877 + -1.5 = 4.84428877 n = 4.84428877 Simplifying n = 4.84428877

Subproblem 2

n + 1.5 = -6.34428877 Simplifying n + 1.5 = -6.34428877 Reorder the terms: 1.5 + n = -6.34428877 Solving 1.5 + n = -6.34428877 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 = -6.34428877 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -6.34428877 + -1.5 n = -6.34428877 + -1.5 Combine like terms: -6.34428877 + -1.5 = -7.84428877 n = -7.84428877 Simplifying n = -7.84428877

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.84428877, -7.84428877}

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