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Simplifying 0 = n2 + 9n + -18 Reorder the terms: 0 = -18 + 9n + n2 Solving 0 = -18 + 9n + n2 Solving for variable 'n'. Combine like terms: 0 + 18 = 18 18 + -9n + -1n2 = -18 + 9n + n2 + 18 + -9n + -1n2 Reorder the terms: 18 + -9n + -1n2 = -18 + 18 + 9n + -9n + n2 + -1n2 Combine like terms: -18 + 18 = 0 18 + -9n + -1n2 = 0 + 9n + -9n + n2 + -1n2 18 + -9n + -1n2 = 9n + -9n + n2 + -1n2 Combine like terms: 9n + -9n = 0 18 + -9n + -1n2 = 0 + n2 + -1n2 18 + -9n + -1n2 = n2 + -1n2 Combine like terms: n2 + -1n2 = 0 18 + -9n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -18 + 9n + n2 = 0 Move the constant term to the right: Add '18' to each side of the equation. -18 + 9n + 18 + n2 = 0 + 18 Reorder the terms: -18 + 18 + 9n + n2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 9n + n2 = 0 + 18 9n + n2 = 0 + 18 Combine like terms: 0 + 18 = 18 9n + n2 = 18 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 = 18 + 20.25 Reorder the terms: 20.25 + 9n + n2 = 18 + 20.25 Combine like terms: 18 + 20.25 = 38.25 20.25 + 9n + n2 = 38.25 Factor a perfect square on the left side: (n + 4.5)(n + 4.5) = 38.25 Calculate the square root of the right side: 6.184658438 Break this problem into two subproblems by setting (n + 4.5) equal to 6.184658438 and -6.184658438.Subproblem 1
n + 4.5 = 6.184658438 Simplifying n + 4.5 = 6.184658438 Reorder the terms: 4.5 + n = 6.184658438 Solving 4.5 + n = 6.184658438 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 = 6.184658438 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 6.184658438 + -4.5 n = 6.184658438 + -4.5 Combine like terms: 6.184658438 + -4.5 = 1.684658438 n = 1.684658438 Simplifying n = 1.684658438Subproblem 2
n + 4.5 = -6.184658438 Simplifying n + 4.5 = -6.184658438 Reorder the terms: 4.5 + n = -6.184658438 Solving 4.5 + n = -6.184658438 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 = -6.184658438 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -6.184658438 + -4.5 n = -6.184658438 + -4.5 Combine like terms: -6.184658438 + -4.5 = -10.684658438 n = -10.684658438 Simplifying n = -10.684658438Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.684658438, -10.684658438}
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