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Simplifying 2n2 + n + -18 = 0 Reorder the terms: -18 + n + 2n2 = 0 Solving -18 + n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -9 + 0.5n + n2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + 0.5n + 9 + n2 = 0 + 9 Reorder the terms: -9 + 9 + 0.5n + n2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + 0.5n + n2 = 0 + 9 0.5n + n2 = 0 + 9 Combine like terms: 0 + 9 = 9 0.5n + n2 = 9 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5n + 0.25 + n2 = 9 + 0.25 Reorder the terms: 0.25 + 0.5n + n2 = 9 + 0.25 Combine like terms: 9 + 0.25 = 9.25 0.25 + 0.5n + n2 = 9.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 9.25 Calculate the square root of the right side: 3.041381265 Break this problem into two subproblems by setting (n + 0.5) equal to 3.041381265 and -3.041381265.Subproblem 1
n + 0.5 = 3.041381265 Simplifying n + 0.5 = 3.041381265 Reorder the terms: 0.5 + n = 3.041381265 Solving 0.5 + n = 3.041381265 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 3.041381265 + -0.5 n = 3.041381265 + -0.5 Combine like terms: 3.041381265 + -0.5 = 2.541381265 n = 2.541381265 Simplifying n = 2.541381265Subproblem 2
n + 0.5 = -3.041381265 Simplifying n + 0.5 = -3.041381265 Reorder the terms: 0.5 + n = -3.041381265 Solving 0.5 + n = -3.041381265 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -3.041381265 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -3.041381265 + -0.5 n = -3.041381265 + -0.5 Combine like terms: -3.041381265 + -0.5 = -3.541381265 n = -3.541381265 Simplifying n = -3.541381265Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.541381265, -3.541381265}
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