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Simplifying 9n + 3n2 + 4 + -0.75 = 180 Reorder the terms: 4 + -0.75 + 9n + 3n2 = 180 Combine like terms: 4 + -0.75 = 3.25 3.25 + 9n + 3n2 = 180 Solving 3.25 + 9n + 3n2 = 180 Solving for variable 'n'. Reorder the terms: 3.25 + -180 + 9n + 3n2 = 180 + -180 Combine like terms: 3.25 + -180 = -176.75 -176.75 + 9n + 3n2 = 180 + -180 Combine like terms: 180 + -180 = 0 -176.75 + 9n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -58.91666667 + 3n + n2 = 0 Move the constant term to the right: Add '58.91666667' to each side of the equation. -58.91666667 + 3n + 58.91666667 + n2 = 0 + 58.91666667 Reorder the terms: -58.91666667 + 58.91666667 + 3n + n2 = 0 + 58.91666667 Combine like terms: -58.91666667 + 58.91666667 = 0.00000000 0.00000000 + 3n + n2 = 0 + 58.91666667 3n + n2 = 0 + 58.91666667 Combine like terms: 0 + 58.91666667 = 58.91666667 3n + n2 = 58.91666667 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 = 58.91666667 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 58.91666667 + 2.25 Combine like terms: 58.91666667 + 2.25 = 61.16666667 2.25 + 3n + n2 = 61.16666667 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 61.16666667 Calculate the square root of the right side: 7.820912138 Break this problem into two subproblems by setting (n + 1.5) equal to 7.820912138 and -7.820912138.Subproblem 1
n + 1.5 = 7.820912138 Simplifying n + 1.5 = 7.820912138 Reorder the terms: 1.5 + n = 7.820912138 Solving 1.5 + n = 7.820912138 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 = 7.820912138 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 7.820912138 + -1.5 n = 7.820912138 + -1.5 Combine like terms: 7.820912138 + -1.5 = 6.320912138 n = 6.320912138 Simplifying n = 6.320912138Subproblem 2
n + 1.5 = -7.820912138 Simplifying n + 1.5 = -7.820912138 Reorder the terms: 1.5 + n = -7.820912138 Solving 1.5 + n = -7.820912138 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 = -7.820912138 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -7.820912138 + -1.5 n = -7.820912138 + -1.5 Combine like terms: -7.820912138 + -1.5 = -9.320912138 n = -9.320912138 Simplifying n = -9.320912138Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.320912138, -9.320912138}
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