n(n+1)+(n+2)=54

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Solution for n(n+1)+(n+2)=54 equation:


Simplifying
n(n + 1) + (n + 2) = 54

Reorder the terms:
n(1 + n) + (n + 2) = 54
(1 * n + n * n) + (n + 2) = 54
(1n + n2) + (n + 2) = 54

Reorder the terms:
1n + n2 + (2 + n) = 54

Remove parenthesis around (2 + n)
1n + n2 + 2 + n = 54

Reorder the terms:
2 + 1n + n + n2 = 54

Combine like terms: 1n + n = 2n
2 + 2n + n2 = 54

Solving
2 + 2n + n2 = 54

Solving for variable 'n'.

Reorder the terms:
2 + -54 + 2n + n2 = 54 + -54

Combine like terms: 2 + -54 = -52
-52 + 2n + n2 = 54 + -54

Combine like terms: 54 + -54 = 0
-52 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '52' to each side of the equation.
-52 + 2n + 52 + n2 = 0 + 52

Reorder the terms:
-52 + 52 + 2n + n2 = 0 + 52

Combine like terms: -52 + 52 = 0
0 + 2n + n2 = 0 + 52
2n + n2 = 0 + 52

Combine like terms: 0 + 52 = 52
2n + n2 = 52

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 52 + 1

Reorder the terms:
1 + 2n + n2 = 52 + 1

Combine like terms: 52 + 1 = 53
1 + 2n + n2 = 53

Factor a perfect square on the left side:
(n + 1)(n + 1) = 53

Calculate the square root of the right side: 7.280109889

Break this problem into two subproblems by setting 
(n + 1) equal to 7.280109889 and -7.280109889.

Subproblem 1

n + 1 = 7.280109889 Simplifying n + 1 = 7.280109889 Reorder the terms: 1 + n = 7.280109889 Solving 1 + n = 7.280109889 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = 7.280109889 + -1 Combine like terms: 1 + -1 = 0 0 + n = 7.280109889 + -1 n = 7.280109889 + -1 Combine like terms: 7.280109889 + -1 = 6.280109889 n = 6.280109889 Simplifying n = 6.280109889

Subproblem 2

n + 1 = -7.280109889 Simplifying n + 1 = -7.280109889 Reorder the terms: 1 + n = -7.280109889 Solving 1 + n = -7.280109889 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + n = -7.280109889 + -1 Combine like terms: 1 + -1 = 0 0 + n = -7.280109889 + -1 n = -7.280109889 + -1 Combine like terms: -7.280109889 + -1 = -8.280109889 n = -8.280109889 Simplifying n = -8.280109889

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.280109889, -8.280109889}

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