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Simplifying 10n2 + 51n + -57500 = 0 Reorder the terms: -57500 + 51n + 10n2 = 0 Solving -57500 + 51n + 10n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -5750 + 5.1n + n2 = 0 Move the constant term to the right: Add '5750' to each side of the equation. -5750 + 5.1n + 5750 + n2 = 0 + 5750 Reorder the terms: -5750 + 5750 + 5.1n + n2 = 0 + 5750 Combine like terms: -5750 + 5750 = 0 0 + 5.1n + n2 = 0 + 5750 5.1n + n2 = 0 + 5750 Combine like terms: 0 + 5750 = 5750 5.1n + n2 = 5750 The n term is 5.1n. Take half its coefficient (2.55). Square it (6.5025) and add it to both sides. Add '6.5025' to each side of the equation. 5.1n + 6.5025 + n2 = 5750 + 6.5025 Reorder the terms: 6.5025 + 5.1n + n2 = 5750 + 6.5025 Combine like terms: 5750 + 6.5025 = 5756.5025 6.5025 + 5.1n + n2 = 5756.5025 Factor a perfect square on the left side: (n + 2.55)(n + 2.55) = 5756.5025 Calculate the square root of the right side: 75.871618541 Break this problem into two subproblems by setting (n + 2.55) equal to 75.871618541 and -75.871618541.Subproblem 1
n + 2.55 = 75.871618541 Simplifying n + 2.55 = 75.871618541 Reorder the terms: 2.55 + n = 75.871618541 Solving 2.55 + n = 75.871618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.55' to each side of the equation. 2.55 + -2.55 + n = 75.871618541 + -2.55 Combine like terms: 2.55 + -2.55 = 0.00 0.00 + n = 75.871618541 + -2.55 n = 75.871618541 + -2.55 Combine like terms: 75.871618541 + -2.55 = 73.321618541 n = 73.321618541 Simplifying n = 73.321618541Subproblem 2
n + 2.55 = -75.871618541 Simplifying n + 2.55 = -75.871618541 Reorder the terms: 2.55 + n = -75.871618541 Solving 2.55 + n = -75.871618541 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.55' to each side of the equation. 2.55 + -2.55 + n = -75.871618541 + -2.55 Combine like terms: 2.55 + -2.55 = 0.00 0.00 + n = -75.871618541 + -2.55 n = -75.871618541 + -2.55 Combine like terms: -75.871618541 + -2.55 = -78.421618541 n = -78.421618541 Simplifying n = -78.421618541Solution
The solution to the problem is based on the solutions from the subproblems. n = {73.321618541, -78.421618541}
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