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Simplifying 8n2 + -23 = 7n Reorder the terms: -23 + 8n2 = 7n Solving -23 + 8n2 = 7n Solving for variable 'n'. Reorder the terms: -23 + -7n + 8n2 = 7n + -7n Combine like terms: 7n + -7n = 0 -23 + -7n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.875 + -0.875n + n2 = 0 Move the constant term to the right: Add '2.875' to each side of the equation. -2.875 + -0.875n + 2.875 + n2 = 0 + 2.875 Reorder the terms: -2.875 + 2.875 + -0.875n + n2 = 0 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + -0.875n + n2 = 0 + 2.875 -0.875n + n2 = 0 + 2.875 Combine like terms: 0 + 2.875 = 2.875 -0.875n + n2 = 2.875 The n term is -0.875n. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875n + 0.19140625 + n2 = 2.875 + 0.19140625 Reorder the terms: 0.19140625 + -0.875n + n2 = 2.875 + 0.19140625 Combine like terms: 2.875 + 0.19140625 = 3.06640625 0.19140625 + -0.875n + n2 = 3.06640625 Factor a perfect square on the left side: (n + -0.4375)(n + -0.4375) = 3.06640625 Calculate the square root of the right side: 1.751115716 Break this problem into two subproblems by setting (n + -0.4375) equal to 1.751115716 and -1.751115716.Subproblem 1
n + -0.4375 = 1.751115716 Simplifying n + -0.4375 = 1.751115716 Reorder the terms: -0.4375 + n = 1.751115716 Solving -0.4375 + n = 1.751115716 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + n = 1.751115716 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + n = 1.751115716 + 0.4375 n = 1.751115716 + 0.4375 Combine like terms: 1.751115716 + 0.4375 = 2.188615716 n = 2.188615716 Simplifying n = 2.188615716Subproblem 2
n + -0.4375 = -1.751115716 Simplifying n + -0.4375 = -1.751115716 Reorder the terms: -0.4375 + n = -1.751115716 Solving -0.4375 + n = -1.751115716 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + n = -1.751115716 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + n = -1.751115716 + 0.4375 n = -1.751115716 + 0.4375 Combine like terms: -1.751115716 + 0.4375 = -1.313615716 n = -1.313615716 Simplifying n = -1.313615716Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.188615716, -1.313615716}
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