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Simplifying n2 + n = 20000 Reorder the terms: n + n2 = 20000 Solving n + n2 = 20000 Solving for variable 'n'. Reorder the terms: -20000 + n + n2 = 20000 + -20000 Combine like terms: 20000 + -20000 = 0 -20000 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '20000' to each side of the equation. -20000 + n + 20000 + n2 = 0 + 20000 Reorder the terms: -20000 + 20000 + n + n2 = 0 + 20000 Combine like terms: -20000 + 20000 = 0 0 + n + n2 = 0 + 20000 n + n2 = 0 + 20000 Combine like terms: 0 + 20000 = 20000 n + n2 = 20000 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. n + 0.25 + n2 = 20000 + 0.25 Reorder the terms: 0.25 + n + n2 = 20000 + 0.25 Combine like terms: 20000 + 0.25 = 20000.25 0.25 + n + n2 = 20000.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 20000.25 Calculate the square root of the right side: 141.422240118 Break this problem into two subproblems by setting (n + 0.5) equal to 141.422240118 and -141.422240118.Subproblem 1
n + 0.5 = 141.422240118 Simplifying n + 0.5 = 141.422240118 Reorder the terms: 0.5 + n = 141.422240118 Solving 0.5 + n = 141.422240118 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 = 141.422240118 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 141.422240118 + -0.5 n = 141.422240118 + -0.5 Combine like terms: 141.422240118 + -0.5 = 140.922240118 n = 140.922240118 Simplifying n = 140.922240118Subproblem 2
n + 0.5 = -141.422240118 Simplifying n + 0.5 = -141.422240118 Reorder the terms: 0.5 + n = -141.422240118 Solving 0.5 + n = -141.422240118 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 = -141.422240118 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -141.422240118 + -0.5 n = -141.422240118 + -0.5 Combine like terms: -141.422240118 + -0.5 = -141.922240118 n = -141.922240118 Simplifying n = -141.922240118Solution
The solution to the problem is based on the solutions from the subproblems. n = {140.922240118, -141.922240118}
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