n*(n+1)=168

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Solution for n*(n+1)=168 equation:


Simplifying
n(n + 1) = 168

Reorder the terms:
n(1 + n) = 168
(1 * n + n * n) = 168
(1n + n2) = 168

Solving
1n + n2 = 168

Solving for variable 'n'.

Reorder the terms:
-168 + 1n + n2 = 168 + -168

Combine like terms: 168 + -168 = 0
-168 + 1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '168' to each side of the equation.
-168 + 1n + 168 + n2 = 0 + 168

Reorder the terms:
-168 + 168 + 1n + n2 = 0 + 168

Combine like terms: -168 + 168 = 0
0 + 1n + n2 = 0 + 168
1n + n2 = 0 + 168

Combine like terms: 0 + 168 = 168
1n + n2 = 168

The n term is 1n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 168 + 0.25

Reorder the terms:
0.25 + n + n2 = 168 + 0.25

Combine like terms: 168 + 0.25 = 168.25
0.25 + n + n2 = 168.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 168.25

Calculate the square root of the right side: 12.971121771

Break this problem into two subproblems by setting 
(n + 0.5) equal to 12.971121771 and -12.971121771.

Subproblem 1

n + 0.5 = 12.971121771 Simplifying n + 0.5 = 12.971121771 Reorder the terms: 0.5 + n = 12.971121771 Solving 0.5 + n = 12.971121771 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 12.971121771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 12.971121771 + -0.5 n = 12.971121771 + -0.5 Combine like terms: 12.971121771 + -0.5 = 12.471121771 n = 12.471121771 Simplifying n = 12.471121771

Subproblem 2

n + 0.5 = -12.971121771 Simplifying n + 0.5 = -12.971121771 Reorder the terms: 0.5 + n = -12.971121771 Solving 0.5 + n = -12.971121771 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -12.971121771 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -12.971121771 + -0.5 n = -12.971121771 + -0.5 Combine like terms: -12.971121771 + -0.5 = -13.471121771 n = -13.471121771 Simplifying n = -13.471121771

Solution

The solution to the problem is based on the solutions from the subproblems. n = {12.471121771, -13.471121771}

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