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Simplifying 64n2 + -126n + -177 = 0 Reorder the terms: -177 + -126n + 64n2 = 0 Solving -177 + -126n + 64n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. -2.765625 + -1.96875n + n2 = 0 Move the constant term to the right: Add '2.765625' to each side of the equation. -2.765625 + -1.96875n + 2.765625 + n2 = 0 + 2.765625 Reorder the terms: -2.765625 + 2.765625 + -1.96875n + n2 = 0 + 2.765625 Combine like terms: -2.765625 + 2.765625 = 0.000000 0.000000 + -1.96875n + n2 = 0 + 2.765625 -1.96875n + n2 = 0 + 2.765625 Combine like terms: 0 + 2.765625 = 2.765625 -1.96875n + n2 = 2.765625 The n term is -1.96875n. Take half its coefficient (-0.984375). Square it (0.9689941406) and add it to both sides. Add '0.9689941406' to each side of the equation. -1.96875n + 0.9689941406 + n2 = 2.765625 + 0.9689941406 Reorder the terms: 0.9689941406 + -1.96875n + n2 = 2.765625 + 0.9689941406 Combine like terms: 2.765625 + 0.9689941406 = 3.7346191406 0.9689941406 + -1.96875n + n2 = 3.7346191406 Factor a perfect square on the left side: (n + -0.984375)(n + -0.984375) = 3.7346191406 Calculate the square root of the right side: 1.932516272 Break this problem into two subproblems by setting (n + -0.984375) equal to 1.932516272 and -1.932516272.Subproblem 1
n + -0.984375 = 1.932516272 Simplifying n + -0.984375 = 1.932516272 Reorder the terms: -0.984375 + n = 1.932516272 Solving -0.984375 + n = 1.932516272 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.984375' to each side of the equation. -0.984375 + 0.984375 + n = 1.932516272 + 0.984375 Combine like terms: -0.984375 + 0.984375 = 0.000000 0.000000 + n = 1.932516272 + 0.984375 n = 1.932516272 + 0.984375 Combine like terms: 1.932516272 + 0.984375 = 2.916891272 n = 2.916891272 Simplifying n = 2.916891272Subproblem 2
n + -0.984375 = -1.932516272 Simplifying n + -0.984375 = -1.932516272 Reorder the terms: -0.984375 + n = -1.932516272 Solving -0.984375 + n = -1.932516272 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.984375' to each side of the equation. -0.984375 + 0.984375 + n = -1.932516272 + 0.984375 Combine like terms: -0.984375 + 0.984375 = 0.000000 0.000000 + n = -1.932516272 + 0.984375 n = -1.932516272 + 0.984375 Combine like terms: -1.932516272 + 0.984375 = -0.948141272 n = -0.948141272 Simplifying n = -0.948141272Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.916891272, -0.948141272}
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