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Simplifying 4 + 88 + -1n * n = 9n Multiply n * n 4 + 88 + -1n2 = 9n Combine like terms: 4 + 88 = 92 92 + -1n2 = 9n Solving 92 + -1n2 = 9n Solving for variable 'n'. Reorder the terms: 92 + -9n + -1n2 = 9n + -9n Combine like terms: 9n + -9n = 0 92 + -9n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -92 + 9n + n2 = 0 Move the constant term to the right: Add '92' to each side of the equation. -92 + 9n + 92 + n2 = 0 + 92 Reorder the terms: -92 + 92 + 9n + n2 = 0 + 92 Combine like terms: -92 + 92 = 0 0 + 9n + n2 = 0 + 92 9n + n2 = 0 + 92 Combine like terms: 0 + 92 = 92 9n + n2 = 92 The n term is 9n. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9n + 20.25 + n2 = 92 + 20.25 Reorder the terms: 20.25 + 9n + n2 = 92 + 20.25 Combine like terms: 92 + 20.25 = 112.25 20.25 + 9n + n2 = 112.25 Factor a perfect square on the left side: (n + 4.5)(n + 4.5) = 112.25 Calculate the square root of the right side: 10.59481005 Break this problem into two subproblems by setting (n + 4.5) equal to 10.59481005 and -10.59481005.Subproblem 1
n + 4.5 = 10.59481005 Simplifying n + 4.5 = 10.59481005 Reorder the terms: 4.5 + n = 10.59481005 Solving 4.5 + n = 10.59481005 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 10.59481005 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 10.59481005 + -4.5 n = 10.59481005 + -4.5 Combine like terms: 10.59481005 + -4.5 = 6.09481005 n = 6.09481005 Simplifying n = 6.09481005Subproblem 2
n + 4.5 = -10.59481005 Simplifying n + 4.5 = -10.59481005 Reorder the terms: 4.5 + n = -10.59481005 Solving 4.5 + n = -10.59481005 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -10.59481005 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -10.59481005 + -4.5 n = -10.59481005 + -4.5 Combine like terms: -10.59481005 + -4.5 = -15.09481005 n = -15.09481005 Simplifying n = -15.09481005Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.09481005, -15.09481005}
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