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Simplifying n2 + -3n + -18 = 11 Reorder the terms: -18 + -3n + n2 = 11 Solving -18 + -3n + n2 = 11 Solving for variable 'n'. Reorder the terms: -18 + -11 + -3n + n2 = 11 + -11 Combine like terms: -18 + -11 = -29 -29 + -3n + n2 = 11 + -11 Combine like terms: 11 + -11 = 0 -29 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '29' to each side of the equation. -29 + -3n + 29 + n2 = 0 + 29 Reorder the terms: -29 + 29 + -3n + n2 = 0 + 29 Combine like terms: -29 + 29 = 0 0 + -3n + n2 = 0 + 29 -3n + n2 = 0 + 29 Combine like terms: 0 + 29 = 29 -3n + n2 = 29 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 29 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 29 + 2.25 Combine like terms: 29 + 2.25 = 31.25 2.25 + -3n + n2 = 31.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 31.25 Calculate the square root of the right side: 5.590169944 Break this problem into two subproblems by setting (n + -1.5) equal to 5.590169944 and -5.590169944.Subproblem 1
n + -1.5 = 5.590169944 Simplifying n + -1.5 = 5.590169944 Reorder the terms: -1.5 + n = 5.590169944 Solving -1.5 + n = 5.590169944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 5.590169944 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 5.590169944 + 1.5 n = 5.590169944 + 1.5 Combine like terms: 5.590169944 + 1.5 = 7.090169944 n = 7.090169944 Simplifying n = 7.090169944Subproblem 2
n + -1.5 = -5.590169944 Simplifying n + -1.5 = -5.590169944 Reorder the terms: -1.5 + n = -5.590169944 Solving -1.5 + n = -5.590169944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -5.590169944 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -5.590169944 + 1.5 n = -5.590169944 + 1.5 Combine like terms: -5.590169944 + 1.5 = -4.090169944 n = -4.090169944 Simplifying n = -4.090169944Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.090169944, -4.090169944}
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