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Simplifying n2 + n + -10.5 = 0 Reorder the terms: -10.5 + n + n2 = 0 Solving -10.5 + n + n2 = 0 Solving for variable 'n'. Begin completing the square. Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + n + 10.5 + n2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + n + n2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + n + n2 = 0 + 10.5 n + n2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 n + n2 = 10.5 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 = 10.5 + 0.25 Reorder the terms: 0.25 + n + n2 = 10.5 + 0.25 Combine like terms: 10.5 + 0.25 = 10.75 0.25 + n + n2 = 10.75 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 10.75 Calculate the square root of the right side: 3.278719262 Break this problem into two subproblems by setting (n + 0.5) equal to 3.278719262 and -3.278719262.Subproblem 1
n + 0.5 = 3.278719262 Simplifying n + 0.5 = 3.278719262 Reorder the terms: 0.5 + n = 3.278719262 Solving 0.5 + n = 3.278719262 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 = 3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 3.278719262 + -0.5 n = 3.278719262 + -0.5 Combine like terms: 3.278719262 + -0.5 = 2.778719262 n = 2.778719262 Simplifying n = 2.778719262Subproblem 2
n + 0.5 = -3.278719262 Simplifying n + 0.5 = -3.278719262 Reorder the terms: 0.5 + n = -3.278719262 Solving 0.5 + n = -3.278719262 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 = -3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -3.278719262 + -0.5 n = -3.278719262 + -0.5 Combine like terms: -3.278719262 + -0.5 = -3.778719262 n = -3.778719262 Simplifying n = -3.778719262Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.778719262, -3.778719262}
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