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Simplifying 2n2 = 9n + -2 Reorder the terms: 2n2 = -2 + 9n Solving 2n2 = -2 + 9n Solving for variable 'n'. Reorder the terms: 2 + -9n + 2n2 = -2 + 9n + 2 + -9n Reorder the terms: 2 + -9n + 2n2 = -2 + 2 + 9n + -9n Combine like terms: -2 + 2 = 0 2 + -9n + 2n2 = 0 + 9n + -9n 2 + -9n + 2n2 = 9n + -9n Combine like terms: 9n + -9n = 0 2 + -9n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1 + -4.5n + n2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -4.5n + -1 + n2 = 0 + -1 Reorder the terms: 1 + -1 + -4.5n + n2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -4.5n + n2 = 0 + -1 -4.5n + n2 = 0 + -1 Combine like terms: 0 + -1 = -1 -4.5n + n2 = -1 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 = -1 + 5.0625 Reorder the terms: 5.0625 + -4.5n + n2 = -1 + 5.0625 Combine like terms: -1 + 5.0625 = 4.0625 5.0625 + -4.5n + n2 = 4.0625 Factor a perfect square on the left side: (n + -2.25)(n + -2.25) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (n + -2.25) equal to 2.015564437 and -2.015564437.Subproblem 1
n + -2.25 = 2.015564437 Simplifying n + -2.25 = 2.015564437 Reorder the terms: -2.25 + n = 2.015564437 Solving -2.25 + n = 2.015564437 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 = 2.015564437 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = 2.015564437 + 2.25 n = 2.015564437 + 2.25 Combine like terms: 2.015564437 + 2.25 = 4.265564437 n = 4.265564437 Simplifying n = 4.265564437Subproblem 2
n + -2.25 = -2.015564437 Simplifying n + -2.25 = -2.015564437 Reorder the terms: -2.25 + n = -2.015564437 Solving -2.25 + n = -2.015564437 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 = -2.015564437 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = -2.015564437 + 2.25 n = -2.015564437 + 2.25 Combine like terms: -2.015564437 + 2.25 = 0.234435563 n = 0.234435563 Simplifying n = 0.234435563Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.265564437, 0.234435563}
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