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Simplifying -28n2 = -96 + -184n Solving -28n2 = -96 + -184n Solving for variable 'n'. Reorder the terms: 96 + 184n + -28n2 = -96 + -184n + 96 + 184n Reorder the terms: 96 + 184n + -28n2 = -96 + 96 + -184n + 184n Combine like terms: -96 + 96 = 0 96 + 184n + -28n2 = 0 + -184n + 184n 96 + 184n + -28n2 = -184n + 184n Combine like terms: -184n + 184n = 0 96 + 184n + -28n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(24 + 46n + -7n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(24 + 46n + -7n2)' equal to zero and attempt to solve: Simplifying 24 + 46n + -7n2 = 0 Solving 24 + 46n + -7n2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -3.428571429 + -6.571428571n + n2 = 0 Move the constant term to the right: Add '3.428571429' to each side of the equation. -3.428571429 + -6.571428571n + 3.428571429 + n2 = 0 + 3.428571429 Reorder the terms: -3.428571429 + 3.428571429 + -6.571428571n + n2 = 0 + 3.428571429 Combine like terms: -3.428571429 + 3.428571429 = 0.000000000 0.000000000 + -6.571428571n + n2 = 0 + 3.428571429 -6.571428571n + n2 = 0 + 3.428571429 Combine like terms: 0 + 3.428571429 = 3.428571429 -6.571428571n + n2 = 3.428571429 The n term is -6.571428571n. Take half its coefficient (-3.285714286). Square it (10.79591837) and add it to both sides. Add '10.79591837' to each side of the equation. -6.571428571n + 10.79591837 + n2 = 3.428571429 + 10.79591837 Reorder the terms: 10.79591837 + -6.571428571n + n2 = 3.428571429 + 10.79591837 Combine like terms: 3.428571429 + 10.79591837 = 14.224489799 10.79591837 + -6.571428571n + n2 = 14.224489799 Factor a perfect square on the left side: (n + -3.285714286)(n + -3.285714286) = 14.224489799 Calculate the square root of the right side: 3.771536795 Break this problem into two subproblems by setting (n + -3.285714286) equal to 3.771536795 and -3.771536795.Subproblem 1
n + -3.285714286 = 3.771536795 Simplifying n + -3.285714286 = 3.771536795 Reorder the terms: -3.285714286 + n = 3.771536795 Solving -3.285714286 + n = 3.771536795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.285714286' to each side of the equation. -3.285714286 + 3.285714286 + n = 3.771536795 + 3.285714286 Combine like terms: -3.285714286 + 3.285714286 = 0.000000000 0.000000000 + n = 3.771536795 + 3.285714286 n = 3.771536795 + 3.285714286 Combine like terms: 3.771536795 + 3.285714286 = 7.057251081 n = 7.057251081 Simplifying n = 7.057251081Subproblem 2
n + -3.285714286 = -3.771536795 Simplifying n + -3.285714286 = -3.771536795 Reorder the terms: -3.285714286 + n = -3.771536795 Solving -3.285714286 + n = -3.771536795 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.285714286' to each side of the equation. -3.285714286 + 3.285714286 + n = -3.771536795 + 3.285714286 Combine like terms: -3.285714286 + 3.285714286 = 0.000000000 0.000000000 + n = -3.771536795 + 3.285714286 n = -3.771536795 + 3.285714286 Combine like terms: -3.771536795 + 3.285714286 = -0.485822509 n = -0.485822509 Simplifying n = -0.485822509Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.057251081, -0.485822509}Solution
n = {7.057251081, -0.485822509}
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