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