171=n^2-n

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Solution for 171=n^2-n equation:


Simplifying
171 = n2 + -1n

Reorder the terms:
171 = -1n + n2

Solving
171 = -1n + n2

Solving for variable 'n'.

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

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

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

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-171 + -1n + n2 = 0

Move the constant term to the right:

Add '171' to each side of the equation.
-171 + -1n + 171 + n2 = 0 + 171

Reorder the terms:
-171 + 171 + -1n + n2 = 0 + 171

Combine like terms: -171 + 171 = 0
0 + -1n + n2 = 0 + 171
-1n + n2 = 0 + 171

Combine like terms: 0 + 171 = 171
-1n + n2 = 171

The n term is n.  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 = 171 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 171 + 0.25

Combine like terms: 171 + 0.25 = 171.25
0.25 + -1n + n2 = 171.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 171.25

Calculate the square root of the right side: 13.086252328

Break this problem into two subproblems by setting 
(n + 0.5) equal to 13.086252328 and -13.086252328.

Subproblem 1

n + 0.5 = 13.086252328 Simplifying n + 0.5 = 13.086252328 Reorder the terms: 0.5 + n = 13.086252328 Solving 0.5 + n = 13.086252328 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 = 13.086252328 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 13.086252328 + -0.5 n = 13.086252328 + -0.5 Combine like terms: 13.086252328 + -0.5 = 12.586252328 n = 12.586252328 Simplifying n = 12.586252328

Subproblem 2

n + 0.5 = -13.086252328 Simplifying n + 0.5 = -13.086252328 Reorder the terms: 0.5 + n = -13.086252328 Solving 0.5 + n = -13.086252328 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 = -13.086252328 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -13.086252328 + -0.5 n = -13.086252328 + -0.5 Combine like terms: -13.086252328 + -0.5 = -13.586252328 n = -13.586252328 Simplifying n = -13.586252328

Solution

The solution to the problem is based on the solutions from the subproblems. n = {12.586252328, -13.586252328}

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