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Simplifying -2n2 + 2n + 46 = 0 Reorder the terms: 46 + 2n + -2n2 = 0 Solving 46 + 2n + -2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(23 + n + -1n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(23 + n + -1n2)' equal to zero and attempt to solve: Simplifying 23 + n + -1n2 = 0 Solving 23 + n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -23 + -1n + n2 = 0 Move the constant term to the right: Add '23' to each side of the equation. -23 + -1n + 23 + n2 = 0 + 23 Reorder the terms: -23 + 23 + -1n + n2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + -1n + n2 = 0 + 23 -1n + n2 = 0 + 23 Combine like terms: 0 + 23 = 23 -1n + n2 = 23 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. -1n + 0.25 + n2 = 23 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 23 + 0.25 Combine like terms: 23 + 0.25 = 23.25 0.25 + -1n + n2 = 23.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 23.25 Calculate the square root of the right side: 4.82182538 Break this problem into two subproblems by setting (n + 0.5) equal to 4.82182538 and -4.82182538.Subproblem 1
n + 0.5 = 4.82182538 Simplifying n + 0.5 = 4.82182538 Reorder the terms: 0.5 + n = 4.82182538 Solving 0.5 + n = 4.82182538 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 = 4.82182538 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 4.82182538 + -0.5 n = 4.82182538 + -0.5 Combine like terms: 4.82182538 + -0.5 = 4.32182538 n = 4.32182538 Simplifying n = 4.32182538Subproblem 2
n + 0.5 = -4.82182538 Simplifying n + 0.5 = -4.82182538 Reorder the terms: 0.5 + n = -4.82182538 Solving 0.5 + n = -4.82182538 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 = -4.82182538 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -4.82182538 + -0.5 n = -4.82182538 + -0.5 Combine like terms: -4.82182538 + -0.5 = -5.32182538 n = -5.32182538 Simplifying n = -5.32182538Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.32182538, -5.32182538}Solution
n = {4.32182538, -5.32182538}
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