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Simplifying 2n2 + 9n + 8 = 0 Reorder the terms: 8 + 9n + 2n2 = 0 Solving 8 + 9n + 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'. 4 + 4.5n + n2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + 4.5n + -4 + n2 = 0 + -4 Reorder the terms: 4 + -4 + 4.5n + n2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + 4.5n + n2 = 0 + -4 4.5n + n2 = 0 + -4 Combine like terms: 0 + -4 = -4 4.5n + n2 = -4 The n term is 4.5n. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5n + 5.0625 + n2 = -4 + 5.0625 Reorder the terms: 5.0625 + 4.5n + n2 = -4 + 5.0625 Combine like terms: -4 + 5.0625 = 1.0625 5.0625 + 4.5n + n2 = 1.0625 Factor a perfect square on the left side: (n + 2.25)(n + 2.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (n + 2.25) equal to 1.030776406 and -1.030776406.Subproblem 1
n + 2.25 = 1.030776406 Simplifying n + 2.25 = 1.030776406 Reorder the terms: 2.25 + n = 1.030776406 Solving 2.25 + n = 1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + n = 1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + n = 1.030776406 + -2.25 n = 1.030776406 + -2.25 Combine like terms: 1.030776406 + -2.25 = -1.219223594 n = -1.219223594 Simplifying n = -1.219223594Subproblem 2
n + 2.25 = -1.030776406 Simplifying n + 2.25 = -1.030776406 Reorder the terms: 2.25 + n = -1.030776406 Solving 2.25 + n = -1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + n = -1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + n = -1.030776406 + -2.25 n = -1.030776406 + -2.25 Combine like terms: -1.030776406 + -2.25 = -3.280776406 n = -3.280776406 Simplifying n = -3.280776406Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.219223594, -3.280776406}
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