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Simplifying 2n2 + 7n + 4 = 0 Reorder the terms: 4 + 7n + 2n2 = 0 Solving 4 + 7n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2 + 3.5n + n2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 3.5n + -2 + n2 = 0 + -2 Reorder the terms: 2 + -2 + 3.5n + n2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 3.5n + n2 = 0 + -2 3.5n + n2 = 0 + -2 Combine like terms: 0 + -2 = -2 3.5n + n2 = -2 The n term is 3.5n. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5n + 3.0625 + n2 = -2 + 3.0625 Reorder the terms: 3.0625 + 3.5n + n2 = -2 + 3.0625 Combine like terms: -2 + 3.0625 = 1.0625 3.0625 + 3.5n + n2 = 1.0625 Factor a perfect square on the left side: (n + 1.75)(n + 1.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (n + 1.75) equal to 1.030776406 and -1.030776406.Subproblem 1
n + 1.75 = 1.030776406 Simplifying n + 1.75 = 1.030776406 Reorder the terms: 1.75 + n = 1.030776406 Solving 1.75 + n = 1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = 1.030776406 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = 1.030776406 + -1.75 n = 1.030776406 + -1.75 Combine like terms: 1.030776406 + -1.75 = -0.719223594 n = -0.719223594 Simplifying n = -0.719223594Subproblem 2
n + 1.75 = -1.030776406 Simplifying n + 1.75 = -1.030776406 Reorder the terms: 1.75 + n = -1.030776406 Solving 1.75 + n = -1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = -1.030776406 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = -1.030776406 + -1.75 n = -1.030776406 + -1.75 Combine like terms: -1.030776406 + -1.75 = -2.780776406 n = -2.780776406 Simplifying n = -2.780776406Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.719223594, -2.780776406}
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