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Simplifying 18n2 + -52n + -81 = 0 Reorder the terms: -81 + -52n + 18n2 = 0 Solving -81 + -52n + 18n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -4.5 + -2.888888889n + n2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + -2.888888889n + 4.5 + n2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + -2.888888889n + n2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + -2.888888889n + n2 = 0 + 4.5 -2.888888889n + n2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 -2.888888889n + n2 = 4.5 The n term is -2.888888889n. Take half its coefficient (-1.444444445). Square it (2.086419755) and add it to both sides. Add '2.086419755' to each side of the equation. -2.888888889n + 2.086419755 + n2 = 4.5 + 2.086419755 Reorder the terms: 2.086419755 + -2.888888889n + n2 = 4.5 + 2.086419755 Combine like terms: 4.5 + 2.086419755 = 6.586419755 2.086419755 + -2.888888889n + n2 = 6.586419755 Factor a perfect square on the left side: (n + -1.444444445)(n + -1.444444445) = 6.586419755 Calculate the square root of the right side: 2.566402103 Break this problem into two subproblems by setting (n + -1.444444445) equal to 2.566402103 and -2.566402103.Subproblem 1
n + -1.444444445 = 2.566402103 Simplifying n + -1.444444445 = 2.566402103 Reorder the terms: -1.444444445 + n = 2.566402103 Solving -1.444444445 + n = 2.566402103 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + n = 2.566402103 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + n = 2.566402103 + 1.444444445 n = 2.566402103 + 1.444444445 Combine like terms: 2.566402103 + 1.444444445 = 4.010846548 n = 4.010846548 Simplifying n = 4.010846548Subproblem 2
n + -1.444444445 = -2.566402103 Simplifying n + -1.444444445 = -2.566402103 Reorder the terms: -1.444444445 + n = -2.566402103 Solving -1.444444445 + n = -2.566402103 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + n = -2.566402103 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + n = -2.566402103 + 1.444444445 n = -2.566402103 + 1.444444445 Combine like terms: -2.566402103 + 1.444444445 = -1.121957658 n = -1.121957658 Simplifying n = -1.121957658Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.010846548, -1.121957658}
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