n*(n-1)=226

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Solution for n*(n-1)=226 equation:


Simplifying
n(n + -1) = 226

Reorder the terms:
n(-1 + n) = 226
(-1 * n + n * n) = 226
(-1n + n2) = 226

Solving
-1n + n2 = 226

Solving for variable 'n'.

Reorder the terms:
-226 + -1n + n2 = 226 + -226

Combine like terms: 226 + -226 = 0
-226 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '226' to each side of the equation.
-226 + -1n + 226 + n2 = 0 + 226

Reorder the terms:
-226 + 226 + -1n + n2 = 0 + 226

Combine like terms: -226 + 226 = 0
0 + -1n + n2 = 0 + 226
-1n + n2 = 0 + 226

Combine like terms: 0 + 226 = 226
-1n + n2 = 226

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.
-1n + 0.25 + n2 = 226 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 226 + 0.25

Combine like terms: 226 + 0.25 = 226.25
0.25 + -1n + n2 = 226.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 226.25

Calculate the square root of the right side: 15.041608956

Break this problem into two subproblems by setting 
(n + -0.5) equal to 15.041608956 and -15.041608956.

Subproblem 1

n + -0.5 = 15.041608956 Simplifying n + -0.5 = 15.041608956 Reorder the terms: -0.5 + n = 15.041608956 Solving -0.5 + n = 15.041608956 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 = 15.041608956 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 15.041608956 + 0.5 n = 15.041608956 + 0.5 Combine like terms: 15.041608956 + 0.5 = 15.541608956 n = 15.541608956 Simplifying n = 15.541608956

Subproblem 2

n + -0.5 = -15.041608956 Simplifying n + -0.5 = -15.041608956 Reorder the terms: -0.5 + n = -15.041608956 Solving -0.5 + n = -15.041608956 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 = -15.041608956 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -15.041608956 + 0.5 n = -15.041608956 + 0.5 Combine like terms: -15.041608956 + 0.5 = -14.541608956 n = -14.541608956 Simplifying n = -14.541608956

Solution

The solution to the problem is based on the solutions from the subproblems. n = {15.541608956, -14.541608956}

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