300=0.5n(n-1)

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Solution for 300=0.5n(n-1) equation:


Simplifying
300 = 0.5n(n + -1)

Reorder the terms:
300 = 0.5n(-1 + n)
300 = (-1 * 0.5n + n * 0.5n)
300 = (-0.5n + 0.5n2)

Solving
300 = -0.5n + 0.5n2

Solving for variable 'n'.

Reorder the terms:
300 + 0.5n + -0.5n2 = -0.5n + 0.5n + 0.5n2 + -0.5n2

Combine like terms: -0.5n + 0.5n = 0.0
300 + 0.5n + -0.5n2 = 0.0 + 0.5n2 + -0.5n2
300 + 0.5n + -0.5n2 = 0.5n2 + -0.5n2

Combine like terms: 0.5n2 + -0.5n2 = 0.0
300 + 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'.
-600 + -1n + n2 = 0

Move the constant term to the right:

Add '600' to each side of the equation.
-600 + -1n + 600 + n2 = 0 + 600

Reorder the terms:
-600 + 600 + -1n + n2 = 0 + 600

Combine like terms: -600 + 600 = 0
0 + -1n + n2 = 0 + 600
-1n + n2 = 0 + 600

Combine like terms: 0 + 600 = 600
-1n + n2 = 600

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 = 600 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 600 + 0.25

Combine like terms: 600 + 0.25 = 600.25
0.25 + -1n + n2 = 600.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 600.25

Calculate the square root of the right side: 24.5

Break this problem into two subproblems by setting 
(n + -0.5) equal to 24.5 and -24.5.

Subproblem 1

n + -0.5 = 24.5 Simplifying n + -0.5 = 24.5 Reorder the terms: -0.5 + n = 24.5 Solving -0.5 + n = 24.5 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 = 24.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 24.5 + 0.5 n = 24.5 + 0.5 Combine like terms: 24.5 + 0.5 = 25 n = 25 Simplifying n = 25

Subproblem 2

n + -0.5 = -24.5 Simplifying n + -0.5 = -24.5 Reorder the terms: -0.5 + n = -24.5 Solving -0.5 + n = -24.5 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 = -24.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -24.5 + 0.5 n = -24.5 + 0.5 Combine like terms: -24.5 + 0.5 = -24 n = -24 Simplifying n = -24

Solution

The solution to the problem is based on the solutions from the subproblems. n = {25, -24}

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