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Simplifying 2n2 + 5n + -3 = 3 Reorder the terms: -3 + 5n + 2n2 = 3 Solving -3 + 5n + 2n2 = 3 Solving for variable 'n'. Reorder the terms: -3 + -3 + 5n + 2n2 = 3 + -3 Combine like terms: -3 + -3 = -6 -6 + 5n + 2n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -6 + 5n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3 + 2.5n + n2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 2.5n + 3 + n2 = 0 + 3 Reorder the terms: -3 + 3 + 2.5n + n2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 2.5n + n2 = 0 + 3 2.5n + n2 = 0 + 3 Combine like terms: 0 + 3 = 3 2.5n + n2 = 3 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 = 3 + 1.5625 Reorder the terms: 1.5625 + 2.5n + n2 = 3 + 1.5625 Combine like terms: 3 + 1.5625 = 4.5625 1.5625 + 2.5n + n2 = 4.5625 Factor a perfect square on the left side: (n + 1.25)(n + 1.25) = 4.5625 Calculate the square root of the right side: 2.136000936 Break this problem into two subproblems by setting (n + 1.25) equal to 2.136000936 and -2.136000936.Subproblem 1
n + 1.25 = 2.136000936 Simplifying n + 1.25 = 2.136000936 Reorder the terms: 1.25 + n = 2.136000936 Solving 1.25 + n = 2.136000936 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 = 2.136000936 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 2.136000936 + -1.25 n = 2.136000936 + -1.25 Combine like terms: 2.136000936 + -1.25 = 0.886000936 n = 0.886000936 Simplifying n = 0.886000936Subproblem 2
n + 1.25 = -2.136000936 Simplifying n + 1.25 = -2.136000936 Reorder the terms: 1.25 + n = -2.136000936 Solving 1.25 + n = -2.136000936 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 = -2.136000936 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -2.136000936 + -1.25 n = -2.136000936 + -1.25 Combine like terms: -2.136000936 + -1.25 = -3.386000936 n = -3.386000936 Simplifying n = -3.386000936Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.886000936, -3.386000936}
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