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Simplifying 2n2 + 5n + 10 = -3 + 12 Reorder the terms: 10 + 5n + 2n2 = -3 + 12 Combine like terms: -3 + 12 = 9 10 + 5n + 2n2 = 9 Solving 10 + 5n + 2n2 = 9 Solving for variable 'n'. Reorder the terms: 10 + -9 + 5n + 2n2 = 9 + -9 Combine like terms: 10 + -9 = 1 1 + 5n + 2n2 = 9 + -9 Combine like terms: 9 + -9 = 0 1 + 5n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 2.5n + n2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 2.5n + -0.5 + n2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 2.5n + n2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 2.5n + n2 = 0 + -0.5 2.5n + n2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 2.5n + n2 = -0.5 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 = -0.5 + 1.5625 Reorder the terms: 1.5625 + 2.5n + n2 = -0.5 + 1.5625 Combine like terms: -0.5 + 1.5625 = 1.0625 1.5625 + 2.5n + n2 = 1.0625 Factor a perfect square on the left side: (n + 1.25)(n + 1.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (n + 1.25) equal to 1.030776406 and -1.030776406.Subproblem 1
n + 1.25 = 1.030776406 Simplifying n + 1.25 = 1.030776406 Reorder the terms: 1.25 + n = 1.030776406 Solving 1.25 + n = 1.030776406 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 = 1.030776406 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 1.030776406 + -1.25 n = 1.030776406 + -1.25 Combine like terms: 1.030776406 + -1.25 = -0.219223594 n = -0.219223594 Simplifying n = -0.219223594Subproblem 2
n + 1.25 = -1.030776406 Simplifying n + 1.25 = -1.030776406 Reorder the terms: 1.25 + n = -1.030776406 Solving 1.25 + n = -1.030776406 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 = -1.030776406 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -1.030776406 + -1.25 n = -1.030776406 + -1.25 Combine like terms: -1.030776406 + -1.25 = -2.280776406 n = -2.280776406 Simplifying n = -2.280776406Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.219223594, -2.280776406}
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