12=0.5n(n-1)

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Solution for 12=0.5n(n-1) equation:


Simplifying
12 = 0.5n(n + -1)

Reorder the terms:
12 = 0.5n(-1 + n)
12 = (-1 * 0.5n + n * 0.5n)
12 = (-0.5n + 0.5n2)

Solving
12 = -0.5n + 0.5n2

Solving for variable 'n'.

Reorder the terms:
12 + 0.5n + -0.5n2 = -0.5n + 0.5n + 0.5n2 + -0.5n2

Combine like terms: -0.5n + 0.5n = 0.0
12 + 0.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2
12 + 0.5n + -0.5n2 = 0.5n2 + -0.5n2

Combine like terms: 0.5n2 + -0.5n2 = 0.0
12 + 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'.
-24 + -1n + n2 = 0

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + -1n + 24 + n2 = 0 + 24

Reorder the terms:
-24 + 24 + -1n + n2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + -1n + n2 = 0 + 24
-1n + n2 = 0 + 24

Combine like terms: 0 + 24 = 24
-1n + n2 = 24

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 = 24 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 24 + 0.25

Combine like terms: 24 + 0.25 = 24.25
0.25 + -1n + n2 = 24.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(n + -0.5) equal to 4.924428901 and -4.924428901.

Subproblem 1

n + -0.5 = 4.924428901 Simplifying n + -0.5 = 4.924428901 Reorder the terms: -0.5 + n = 4.924428901 Solving -0.5 + n = 4.924428901 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 = 4.924428901 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 4.924428901 + 0.5 n = 4.924428901 + 0.5 Combine like terms: 4.924428901 + 0.5 = 5.424428901 n = 5.424428901 Simplifying n = 5.424428901

Subproblem 2

n + -0.5 = -4.924428901 Simplifying n + -0.5 = -4.924428901 Reorder the terms: -0.5 + n = -4.924428901 Solving -0.5 + n = -4.924428901 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 = -4.924428901 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -4.924428901 + 0.5 n = -4.924428901 + 0.5 Combine like terms: -4.924428901 + 0.5 = -4.424428901 n = -4.424428901 Simplifying n = -4.424428901

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.424428901, -4.424428901}

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