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Simplifying 28n2 + 128n + 96 = 0 Reorder the terms: 96 + 128n + 28n2 = 0 Solving 96 + 128n + 28n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4'. 4(24 + 32n + 7n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(24 + 32n + 7n2)' equal to zero and attempt to solve: Simplifying 24 + 32n + 7n2 = 0 Solving 24 + 32n + 7n2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 3.428571429 + 4.571428571n + n2 = 0 Move the constant term to the right: Add '-3.428571429' to each side of the equation. 3.428571429 + 4.571428571n + -3.428571429 + n2 = 0 + -3.428571429 Reorder the terms: 3.428571429 + -3.428571429 + 4.571428571n + n2 = 0 + -3.428571429 Combine like terms: 3.428571429 + -3.428571429 = 0.000000000 0.000000000 + 4.571428571n + n2 = 0 + -3.428571429 4.571428571n + n2 = 0 + -3.428571429 Combine like terms: 0 + -3.428571429 = -3.428571429 4.571428571n + n2 = -3.428571429 The n term is 4.571428571n. Take half its coefficient (2.285714286). Square it (5.224489797) and add it to both sides. Add '5.224489797' to each side of the equation. 4.571428571n + 5.224489797 + n2 = -3.428571429 + 5.224489797 Reorder the terms: 5.224489797 + 4.571428571n + n2 = -3.428571429 + 5.224489797 Combine like terms: -3.428571429 + 5.224489797 = 1.795918368 5.224489797 + 4.571428571n + n2 = 1.795918368 Factor a perfect square on the left side: (n + 2.285714286)(n + 2.285714286) = 1.795918368 Calculate the square root of the right side: 1.340118789 Break this problem into two subproblems by setting (n + 2.285714286) equal to 1.340118789 and -1.340118789.Subproblem 1
n + 2.285714286 = 1.340118789 Simplifying n + 2.285714286 = 1.340118789 Reorder the terms: 2.285714286 + n = 1.340118789 Solving 2.285714286 + n = 1.340118789 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + n = 1.340118789 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + n = 1.340118789 + -2.285714286 n = 1.340118789 + -2.285714286 Combine like terms: 1.340118789 + -2.285714286 = -0.945595497 n = -0.945595497 Simplifying n = -0.945595497Subproblem 2
n + 2.285714286 = -1.340118789 Simplifying n + 2.285714286 = -1.340118789 Reorder the terms: 2.285714286 + n = -1.340118789 Solving 2.285714286 + n = -1.340118789 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + n = -1.340118789 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + n = -1.340118789 + -2.285714286 n = -1.340118789 + -2.285714286 Combine like terms: -1.340118789 + -2.285714286 = -3.625833075 n = -3.625833075 Simplifying n = -3.625833075Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.945595497, -3.625833075}Solution
n = {-0.945595497, -3.625833075}
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