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Simplifying 0 = 2n2 + 2n + -1000 Reorder the terms: 0 = -1000 + 2n + 2n2 Solving 0 = -1000 + 2n + 2n2 Solving for variable 'n'. Combine like terms: 0 + 1000 = 1000 1000 + -2n + -2n2 = -1000 + 2n + 2n2 + 1000 + -2n + -2n2 Reorder the terms: 1000 + -2n + -2n2 = -1000 + 1000 + 2n + -2n + 2n2 + -2n2 Combine like terms: -1000 + 1000 = 0 1000 + -2n + -2n2 = 0 + 2n + -2n + 2n2 + -2n2 1000 + -2n + -2n2 = 2n + -2n + 2n2 + -2n2 Combine like terms: 2n + -2n = 0 1000 + -2n + -2n2 = 0 + 2n2 + -2n2 1000 + -2n + -2n2 = 2n2 + -2n2 Combine like terms: 2n2 + -2n2 = 0 1000 + -2n + -2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(500 + -1n + -1n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(500 + -1n + -1n2)' equal to zero and attempt to solve: Simplifying 500 + -1n + -1n2 = 0 Solving 500 + -1n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -500 + n + n2 = 0 Move the constant term to the right: Add '500' to each side of the equation. -500 + n + 500 + n2 = 0 + 500 Reorder the terms: -500 + 500 + n + n2 = 0 + 500 Combine like terms: -500 + 500 = 0 0 + n + n2 = 0 + 500 n + n2 = 0 + 500 Combine like terms: 0 + 500 = 500 n + n2 = 500 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 = 500 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = 500 + 0.25 Combine like terms: 500 + 0.25 = 500.25 1.25 + n2 = 500.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 500.25 Calculate the square root of the right side: 22.366269246 Break this problem into two subproblems by setting (n + 0.5) equal to 22.366269246 and -22.366269246.Subproblem 1
n + 0.5 = 22.366269246 Simplifying n + 0.5 = 22.366269246 Reorder the terms: 0.5 + n = 22.366269246 Solving 0.5 + n = 22.366269246 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 = 22.366269246 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 22.366269246 + -0.5 n = 22.366269246 + -0.5 Combine like terms: 22.366269246 + -0.5 = 21.866269246 n = 21.866269246 Simplifying n = 21.866269246Subproblem 2
n + 0.5 = -22.366269246 Simplifying n + 0.5 = -22.366269246 Reorder the terms: 0.5 + n = -22.366269246 Solving 0.5 + n = -22.366269246 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 = -22.366269246 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -22.366269246 + -0.5 n = -22.366269246 + -0.5 Combine like terms: -22.366269246 + -0.5 = -22.866269246 n = -22.866269246 Simplifying n = -22.866269246Solution
The solution to the problem is based on the solutions from the subproblems. n = {21.866269246, -22.866269246}Solution
n = {21.866269246, -22.866269246}
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