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

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


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

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

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

Multiply (2 + n) * (4 + n)
(2(4 + n) + n(4 + n)) = 42
((4 * 2 + n * 2) + n(4 + n)) = 42
((8 + 2n) + n(4 + n)) = 42
(8 + 2n + (4 * n + n * n)) = 42
(8 + 2n + (4n + n2)) = 42

Combine like terms: 2n + 4n = 6n
(8 + 6n + n2) = 42

Solving
8 + 6n + n2 = 42

Solving for variable 'n'.

Reorder the terms:
8 + -42 + 6n + n2 = 42 + -42

Combine like terms: 8 + -42 = -34
-34 + 6n + n2 = 42 + -42

Combine like terms: 42 + -42 = 0
-34 + 6n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + 6n + 34 + n2 = 0 + 34

Reorder the terms:
-34 + 34 + 6n + n2 = 0 + 34

Combine like terms: -34 + 34 = 0
0 + 6n + n2 = 0 + 34
6n + n2 = 0 + 34

Combine like terms: 0 + 34 = 34
6n + n2 = 34

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 = 34 + 9

Reorder the terms:
9 + 6n + n2 = 34 + 9

Combine like terms: 34 + 9 = 43
9 + 6n + n2 = 43

Factor a perfect square on the left side:
(n + 3)(n + 3) = 43

Calculate the square root of the right side: 6.557438524

Break this problem into two subproblems by setting 
(n + 3) equal to 6.557438524 and -6.557438524.

Subproblem 1

n + 3 = 6.557438524 Simplifying n + 3 = 6.557438524 Reorder the terms: 3 + n = 6.557438524 Solving 3 + n = 6.557438524 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 = 6.557438524 + -3 Combine like terms: 3 + -3 = 0 0 + n = 6.557438524 + -3 n = 6.557438524 + -3 Combine like terms: 6.557438524 + -3 = 3.557438524 n = 3.557438524 Simplifying n = 3.557438524

Subproblem 2

n + 3 = -6.557438524 Simplifying n + 3 = -6.557438524 Reorder the terms: 3 + n = -6.557438524 Solving 3 + n = -6.557438524 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 = -6.557438524 + -3 Combine like terms: 3 + -3 = 0 0 + n = -6.557438524 + -3 n = -6.557438524 + -3 Combine like terms: -6.557438524 + -3 = -9.557438524 n = -9.557438524 Simplifying n = -9.557438524

Solution

The solution to the problem is based on the solutions from the subproblems. n = {3.557438524, -9.557438524}

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