n^2-n=208

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Solution for n^2-n=208 equation:


Simplifying
n2 + -1n = 208

Reorder the terms:
-1n + n2 = 208

Solving
-1n + n2 = 208

Solving for variable 'n'.

Reorder the terms:
-208 + -1n + n2 = 208 + -208

Combine like terms: 208 + -208 = 0
-208 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '208' to each side of the equation.
-208 + -1n + 208 + n2 = 0 + 208

Reorder the terms:
-208 + 208 + -1n + n2 = 0 + 208

Combine like terms: -208 + 208 = 0
0 + -1n + n2 = 0 + 208
-1n + n2 = 0 + 208

Combine like terms: 0 + 208 = 208
-1n + n2 = 208

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 = 208 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 208 + 0.25

Combine like terms: 208 + 0.25 = 208.25
0.25 + -1n + n2 = 208.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 208.25

Calculate the square root of the right side: 14.43086969

Break this problem into two subproblems by setting 
(n + -0.5) equal to 14.43086969 and -14.43086969.

Subproblem 1

n + -0.5 = 14.43086969 Simplifying n + -0.5 = 14.43086969 Reorder the terms: -0.5 + n = 14.43086969 Solving -0.5 + n = 14.43086969 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 = 14.43086969 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 14.43086969 + 0.5 n = 14.43086969 + 0.5 Combine like terms: 14.43086969 + 0.5 = 14.93086969 n = 14.93086969 Simplifying n = 14.93086969

Subproblem 2

n + -0.5 = -14.43086969 Simplifying n + -0.5 = -14.43086969 Reorder the terms: -0.5 + n = -14.43086969 Solving -0.5 + n = -14.43086969 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 = -14.43086969 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -14.43086969 + 0.5 n = -14.43086969 + 0.5 Combine like terms: -14.43086969 + 0.5 = -13.93086969 n = -13.93086969 Simplifying n = -13.93086969

Solution

The solution to the problem is based on the solutions from the subproblems. n = {14.93086969, -13.93086969}

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