0.5n(n+1)=66

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Solution for 0.5n(n+1)=66 equation:


Simplifying
0.5n(n + 1) = 66

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

Solving
0.5n + 0.5n2 = 66

Solving for variable 'n'.

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

Combine like terms: 66 + -66 = 0
-66 + 0.5n + 0.5n2 = 0

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

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

Move the constant term to the right:

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

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

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

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

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 = 132 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 132 + 0.25

Combine like terms: 132 + 0.25 = 132.25
1.25 + 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 = 11 n = 11 Simplifying n = 11

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 = -12 n = -12 Simplifying n = -12

Solution

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

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