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Simplifying 2n2 = 16 + 9n Solving 2n2 = 16 + 9n Solving for variable 'n'. Reorder the terms: -16 + -9n + 2n2 = 16 + 9n + -16 + -9n Reorder the terms: -16 + -9n + 2n2 = 16 + -16 + 9n + -9n Combine like terms: 16 + -16 = 0 -16 + -9n + 2n2 = 0 + 9n + -9n -16 + -9n + 2n2 = 9n + -9n Combine like terms: 9n + -9n = 0 -16 + -9n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8 + -4.5n + n2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -4.5n + 8 + n2 = 0 + 8 Reorder the terms: -8 + 8 + -4.5n + n2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -4.5n + n2 = 0 + 8 -4.5n + n2 = 0 + 8 Combine like terms: 0 + 8 = 8 -4.5n + n2 = 8 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 = 8 + 5.0625 Reorder the terms: 5.0625 + -4.5n + n2 = 8 + 5.0625 Combine like terms: 8 + 5.0625 = 13.0625 5.0625 + -4.5n + n2 = 13.0625 Factor a perfect square on the left side: (n + -2.25)(n + -2.25) = 13.0625 Calculate the square root of the right side: 3.614208074 Break this problem into two subproblems by setting (n + -2.25) equal to 3.614208074 and -3.614208074.Subproblem 1
n + -2.25 = 3.614208074 Simplifying n + -2.25 = 3.614208074 Reorder the terms: -2.25 + n = 3.614208074 Solving -2.25 + n = 3.614208074 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 = 3.614208074 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = 3.614208074 + 2.25 n = 3.614208074 + 2.25 Combine like terms: 3.614208074 + 2.25 = 5.864208074 n = 5.864208074 Simplifying n = 5.864208074Subproblem 2
n + -2.25 = -3.614208074 Simplifying n + -2.25 = -3.614208074 Reorder the terms: -2.25 + n = -3.614208074 Solving -2.25 + n = -3.614208074 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 = -3.614208074 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = -3.614208074 + 2.25 n = -3.614208074 + 2.25 Combine like terms: -3.614208074 + 2.25 = -1.364208074 n = -1.364208074 Simplifying n = -1.364208074Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.864208074, -1.364208074}
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