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Simplifying n2 + -3n + -28 = 1600 Reorder the terms: -28 + -3n + n2 = 1600 Solving -28 + -3n + n2 = 1600 Solving for variable 'n'. Reorder the terms: -28 + -1600 + -3n + n2 = 1600 + -1600 Combine like terms: -28 + -1600 = -1628 -1628 + -3n + n2 = 1600 + -1600 Combine like terms: 1600 + -1600 = 0 -1628 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '1628' to each side of the equation. -1628 + -3n + 1628 + n2 = 0 + 1628 Reorder the terms: -1628 + 1628 + -3n + n2 = 0 + 1628 Combine like terms: -1628 + 1628 = 0 0 + -3n + n2 = 0 + 1628 -3n + n2 = 0 + 1628 Combine like terms: 0 + 1628 = 1628 -3n + n2 = 1628 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 1628 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 1628 + 2.25 Combine like terms: 1628 + 2.25 = 1630.25 2.25 + -3n + n2 = 1630.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 1630.25 Calculate the square root of the right side: 40.376354466 Break this problem into two subproblems by setting (n + -1.5) equal to 40.376354466 and -40.376354466.Subproblem 1
n + -1.5 = 40.376354466 Simplifying n + -1.5 = 40.376354466 Reorder the terms: -1.5 + n = 40.376354466 Solving -1.5 + n = 40.376354466 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 40.376354466 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 40.376354466 + 1.5 n = 40.376354466 + 1.5 Combine like terms: 40.376354466 + 1.5 = 41.876354466 n = 41.876354466 Simplifying n = 41.876354466Subproblem 2
n + -1.5 = -40.376354466 Simplifying n + -1.5 = -40.376354466 Reorder the terms: -1.5 + n = -40.376354466 Solving -1.5 + n = -40.376354466 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -40.376354466 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -40.376354466 + 1.5 n = -40.376354466 + 1.5 Combine like terms: -40.376354466 + 1.5 = -38.876354466 n = -38.876354466 Simplifying n = -38.876354466Solution
The solution to the problem is based on the solutions from the subproblems. n = {41.876354466, -38.876354466}
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