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Simplifying 64n2 + 56n + -56 = 0 Reorder the terms: -56 + 56n + 64n2 = 0 Solving -56 + 56n + 64n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '8'. 8(-7 + 7n + 8n2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-7 + 7n + 8n2)' equal to zero and attempt to solve: Simplifying -7 + 7n + 8n2 = 0 Solving -7 + 7n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.875 + 0.875n + n2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + 0.875n + 0.875 + n2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + 0.875n + n2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + 0.875n + n2 = 0 + 0.875 0.875n + n2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 0.875n + n2 = 0.875 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 = 0.875 + 0.19140625 Reorder the terms: 0.19140625 + 0.875n + n2 = 0.875 + 0.19140625 Combine like terms: 0.875 + 0.19140625 = 1.06640625 0.19140625 + 0.875n + n2 = 1.06640625 Factor a perfect square on the left side: (n + 0.4375)(n + 0.4375) = 1.06640625 Calculate the square root of the right side: 1.032669478 Break this problem into two subproblems by setting (n + 0.4375) equal to 1.032669478 and -1.032669478.Subproblem 1
n + 0.4375 = 1.032669478 Simplifying n + 0.4375 = 1.032669478 Reorder the terms: 0.4375 + n = 1.032669478 Solving 0.4375 + n = 1.032669478 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 = 1.032669478 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + n = 1.032669478 + -0.4375 n = 1.032669478 + -0.4375 Combine like terms: 1.032669478 + -0.4375 = 0.595169478 n = 0.595169478 Simplifying n = 0.595169478Subproblem 2
n + 0.4375 = -1.032669478 Simplifying n + 0.4375 = -1.032669478 Reorder the terms: 0.4375 + n = -1.032669478 Solving 0.4375 + n = -1.032669478 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 = -1.032669478 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + n = -1.032669478 + -0.4375 n = -1.032669478 + -0.4375 Combine like terms: -1.032669478 + -0.4375 = -1.470169478 n = -1.470169478 Simplifying n = -1.470169478Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.595169478, -1.470169478}Solution
n = {0.595169478, -1.470169478}
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