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Simplifying 28n2 = -96 + -114n Solving 28n2 = -96 + -114n Solving for variable 'n'. Reorder the terms: 96 + 114n + 28n2 = -96 + -114n + 96 + 114n Reorder the terms: 96 + 114n + 28n2 = -96 + 96 + -114n + 114n Combine like terms: -96 + 96 = 0 96 + 114n + 28n2 = 0 + -114n + 114n 96 + 114n + 28n2 = -114n + 114n Combine like terms: -114n + 114n = 0 96 + 114n + 28n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(48 + 57n + 14n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(48 + 57n + 14n2)' equal to zero and attempt to solve: Simplifying 48 + 57n + 14n2 = 0 Solving 48 + 57n + 14n2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. 3.428571429 + 4.071428571n + n2 = 0 Move the constant term to the right: Add '-3.428571429' to each side of the equation. 3.428571429 + 4.071428571n + -3.428571429 + n2 = 0 + -3.428571429 Reorder the terms: 3.428571429 + -3.428571429 + 4.071428571n + n2 = 0 + -3.428571429 Combine like terms: 3.428571429 + -3.428571429 = 0.000000000 0.000000000 + 4.071428571n + n2 = 0 + -3.428571429 4.071428571n + n2 = 0 + -3.428571429 Combine like terms: 0 + -3.428571429 = -3.428571429 4.071428571n + n2 = -3.428571429 The n term is 4.071428571n. Take half its coefficient (2.035714286). Square it (4.144132654) and add it to both sides. Add '4.144132654' to each side of the equation. 4.071428571n + 4.144132654 + n2 = -3.428571429 + 4.144132654 Reorder the terms: 4.144132654 + 4.071428571n + n2 = -3.428571429 + 4.144132654 Combine like terms: -3.428571429 + 4.144132654 = 0.715561225 4.144132654 + 4.071428571n + n2 = 0.715561225 Factor a perfect square on the left side: (n + 2.035714286)(n + 2.035714286) = 0.715561225 Calculate the square root of the right side: 0.84590852 Break this problem into two subproblems by setting (n + 2.035714286) equal to 0.84590852 and -0.84590852.Subproblem 1
n + 2.035714286 = 0.84590852 Simplifying n + 2.035714286 = 0.84590852 Reorder the terms: 2.035714286 + n = 0.84590852 Solving 2.035714286 + n = 0.84590852 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.035714286' to each side of the equation. 2.035714286 + -2.035714286 + n = 0.84590852 + -2.035714286 Combine like terms: 2.035714286 + -2.035714286 = 0.000000000 0.000000000 + n = 0.84590852 + -2.035714286 n = 0.84590852 + -2.035714286 Combine like terms: 0.84590852 + -2.035714286 = -1.189805766 n = -1.189805766 Simplifying n = -1.189805766Subproblem 2
n + 2.035714286 = -0.84590852 Simplifying n + 2.035714286 = -0.84590852 Reorder the terms: 2.035714286 + n = -0.84590852 Solving 2.035714286 + n = -0.84590852 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.035714286' to each side of the equation. 2.035714286 + -2.035714286 + n = -0.84590852 + -2.035714286 Combine like terms: 2.035714286 + -2.035714286 = 0.000000000 0.000000000 + n = -0.84590852 + -2.035714286 n = -0.84590852 + -2.035714286 Combine like terms: -0.84590852 + -2.035714286 = -2.881622806 n = -2.881622806 Simplifying n = -2.881622806Solution
The solution to the problem is based on the solutions from the subproblems. n = {-1.189805766, -2.881622806}Solution
n = {-1.189805766, -2.881622806}
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