.5n(n+1)-188=0

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Solution for .5n(n+1)-188=0 equation:


Simplifying
0.5n(n + 1) + -188 = 0

Reorder the terms:
0.5n(1 + n) + -188 = 0
(1 * 0.5n + n * 0.5n) + -188 = 0
(0.5n + 0.5n2) + -188 = 0

Reorder the terms:
-188 + 0.5n + 0.5n2 = 0

Solving
-188 + 0.5n + 0.5n2 = 0

Solving for variable 'n'.

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

Divide each side by '0.5'.
-376 + n + n2 = 0

Move the constant term to the right:

Add '376' to each side of the equation.
-376 + n + 376 + n2 = 0 + 376

Reorder the terms:
-376 + 376 + n + n2 = 0 + 376

Combine like terms: -376 + 376 = 0
0 + n + n2 = 0 + 376
n + n2 = 0 + 376

Combine like terms: 0 + 376 = 376
n + n2 = 376

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.
 + 0.25 + n2 = 376 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 376 + 0.25

Combine like terms: 376 + 0.25 = 376.25
1.25 + n2 = 376.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 376.25

Calculate the square root of the right side: 19.397164741

Break this problem into two subproblems by setting 
(n + 0.5) equal to 19.397164741 and -19.397164741.

Subproblem 1

n + 0.5 = 19.397164741 Simplifying n + 0.5 = 19.397164741 Reorder the terms: 0.5 + n = 19.397164741 Solving 0.5 + n = 19.397164741 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 = 19.397164741 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 19.397164741 + -0.5 n = 19.397164741 + -0.5 Combine like terms: 19.397164741 + -0.5 = 18.897164741 n = 18.897164741 Simplifying n = 18.897164741

Subproblem 2

n + 0.5 = -19.397164741 Simplifying n + 0.5 = -19.397164741 Reorder the terms: 0.5 + n = -19.397164741 Solving 0.5 + n = -19.397164741 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 = -19.397164741 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -19.397164741 + -0.5 n = -19.397164741 + -0.5 Combine like terms: -19.397164741 + -0.5 = -19.897164741 n = -19.897164741 Simplifying n = -19.897164741

Solution

The solution to the problem is based on the solutions from the subproblems. n = {18.897164741, -19.897164741}

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