66=0.5(n^2-n)

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Solution for 66=0.5(n^2-n) equation:


Simplifying
66 = 0.5(n2 + -1n)

Reorder the terms:
66 = 0.5(-1n + n2)
66 = (-1n * 0.5 + n2 * 0.5)
66 = (-0.5n + 0.5n2)

Solving
66 = -0.5n + 0.5n2

Solving for variable 'n'.

Reorder the terms:
66 + 0.5n + -0.5n2 = -0.5n + 0.5n + 0.5n2 + -0.5n2

Combine like terms: -0.5n + 0.5n = 0.0
66 + 0.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2
66 + 0.5n + -0.5n2 = 0.5n2 + -0.5n2

Combine like terms: 0.5n2 + -0.5n2 = 0.0
66 + 0.5n + -0.5n2 = 0.0

Begin completing the square.  Divide all terms by
-0.5 the coefficient of the squared term: 

Divide each side by '-0.5'.
-132 + -1n + n2 = 0

Move the constant term to the right:

Add '132' to each side of the equation.
-132 + -1n + 132 + n2 = 0 + 132

Reorder the terms:
-132 + 132 + -1n + n2 = 0 + 132

Combine like terms: -132 + 132 = 0
0 + -1n + n2 = 0 + 132
-1n + n2 = 0 + 132

Combine like terms: 0 + 132 = 132
-1n + n2 = 132

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 = 132 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 132 + 0.25

Combine like terms: 132 + 0.25 = 132.25
0.25 + -1n + n2 = 132.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 132.25

Calculate the square root of the right side: 11.5

Break this problem into two subproblems by setting 
(n + -0.5) equal to 11.5 and -11.5.

Subproblem 1

n + -0.5 = 11.5 Simplifying n + -0.5 = 11.5 Reorder the terms: -0.5 + n = 11.5 Solving -0.5 + n = 11.5 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 = 11.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 11.5 + 0.5 n = 11.5 + 0.5 Combine like terms: 11.5 + 0.5 = 12 n = 12 Simplifying n = 12

Subproblem 2

n + -0.5 = -11.5 Simplifying n + -0.5 = -11.5 Reorder the terms: -0.5 + n = -11.5 Solving -0.5 + n = -11.5 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 = -11.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -11.5 + 0.5 n = -11.5 + 0.5 Combine like terms: -11.5 + 0.5 = -11 n = -11 Simplifying n = -11

Solution

The solution to the problem is based on the solutions from the subproblems. n = {12, -11}

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