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Simplifying 2n2 + 5n = 24 Reorder the terms: 5n + 2n2 = 24 Solving 5n + 2n2 = 24 Solving for variable 'n'. Reorder the terms: -24 + 5n + 2n2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 5n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -12 + 2.5n + n2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 2.5n + 12 + n2 = 0 + 12 Reorder the terms: -12 + 12 + 2.5n + n2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 2.5n + n2 = 0 + 12 2.5n + n2 = 0 + 12 Combine like terms: 0 + 12 = 12 2.5n + n2 = 12 The n term is 2.5n. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5n + 1.5625 + n2 = 12 + 1.5625 Reorder the terms: 1.5625 + 2.5n + n2 = 12 + 1.5625 Combine like terms: 12 + 1.5625 = 13.5625 1.5625 + 2.5n + n2 = 13.5625 Factor a perfect square on the left side: (n + 1.25)(n + 1.25) = 13.5625 Calculate the square root of the right side: 3.682729966 Break this problem into two subproblems by setting (n + 1.25) equal to 3.682729966 and -3.682729966.Subproblem 1
n + 1.25 = 3.682729966 Simplifying n + 1.25 = 3.682729966 Reorder the terms: 1.25 + n = 3.682729966 Solving 1.25 + n = 3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = 3.682729966 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 3.682729966 + -1.25 n = 3.682729966 + -1.25 Combine like terms: 3.682729966 + -1.25 = 2.432729966 n = 2.432729966 Simplifying n = 2.432729966Subproblem 2
n + 1.25 = -3.682729966 Simplifying n + 1.25 = -3.682729966 Reorder the terms: 1.25 + n = -3.682729966 Solving 1.25 + n = -3.682729966 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + n = -3.682729966 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -3.682729966 + -1.25 n = -3.682729966 + -1.25 Combine like terms: -3.682729966 + -1.25 = -4.932729966 n = -4.932729966 Simplifying n = -4.932729966Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.432729966, -4.932729966}
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