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Simplifying n2 + n + -512 = 0 Reorder the terms: -512 + n + n2 = 0 Solving -512 + n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '512' to each side of the equation. -512 + n + 512 + n2 = 0 + 512 Reorder the terms: -512 + 512 + n + n2 = 0 + 512 Combine like terms: -512 + 512 = 0 0 + n + n2 = 0 + 512 n + n2 = 0 + 512 Combine like terms: 0 + 512 = 512 n + n2 = 512 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 = 512 + 0.25 Reorder the terms: 0.25 + n + n2 = 512 + 0.25 Combine like terms: 512 + 0.25 = 512.25 0.25 + n + n2 = 512.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 512.25 Calculate the square root of the right side: 22.632940596 Break this problem into two subproblems by setting (n + 0.5) equal to 22.632940596 and -22.632940596.Subproblem 1
n + 0.5 = 22.632940596 Simplifying n + 0.5 = 22.632940596 Reorder the terms: 0.5 + n = 22.632940596 Solving 0.5 + n = 22.632940596 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 = 22.632940596 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 22.632940596 + -0.5 n = 22.632940596 + -0.5 Combine like terms: 22.632940596 + -0.5 = 22.132940596 n = 22.132940596 Simplifying n = 22.132940596Subproblem 2
n + 0.5 = -22.632940596 Simplifying n + 0.5 = -22.632940596 Reorder the terms: 0.5 + n = -22.632940596 Solving 0.5 + n = -22.632940596 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 = -22.632940596 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -22.632940596 + -0.5 n = -22.632940596 + -0.5 Combine like terms: -22.632940596 + -0.5 = -23.132940596 n = -23.132940596 Simplifying n = -23.132940596Solution
The solution to the problem is based on the solutions from the subproblems. n = {22.132940596, -23.132940596}
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