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Simplifying 8n2 + -6n + -21 = 0 Reorder the terms: -21 + -6n + 8n2 = 0 Solving -21 + -6n + 8n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.625 + -0.75n + n2 = 0 Move the constant term to the right: Add '2.625' to each side of the equation. -2.625 + -0.75n + 2.625 + n2 = 0 + 2.625 Reorder the terms: -2.625 + 2.625 + -0.75n + n2 = 0 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + -0.75n + n2 = 0 + 2.625 -0.75n + n2 = 0 + 2.625 Combine like terms: 0 + 2.625 = 2.625 -0.75n + n2 = 2.625 The n term is -0.75n. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75n + 0.140625 + n2 = 2.625 + 0.140625 Reorder the terms: 0.140625 + -0.75n + n2 = 2.625 + 0.140625 Combine like terms: 2.625 + 0.140625 = 2.765625 0.140625 + -0.75n + n2 = 2.765625 Factor a perfect square on the left side: (n + -0.375)(n + -0.375) = 2.765625 Calculate the square root of the right side: 1.663016837 Break this problem into two subproblems by setting (n + -0.375) equal to 1.663016837 and -1.663016837.Subproblem 1
n + -0.375 = 1.663016837 Simplifying n + -0.375 = 1.663016837 Reorder the terms: -0.375 + n = 1.663016837 Solving -0.375 + n = 1.663016837 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = 1.663016837 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 1.663016837 + 0.375 n = 1.663016837 + 0.375 Combine like terms: 1.663016837 + 0.375 = 2.038016837 n = 2.038016837 Simplifying n = 2.038016837Subproblem 2
n + -0.375 = -1.663016837 Simplifying n + -0.375 = -1.663016837 Reorder the terms: -0.375 + n = -1.663016837 Solving -0.375 + n = -1.663016837 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = -1.663016837 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -1.663016837 + 0.375 n = -1.663016837 + 0.375 Combine like terms: -1.663016837 + 0.375 = -1.288016837 n = -1.288016837 Simplifying n = -1.288016837Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.038016837, -1.288016837}
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