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Simplifying 56n3 + 24n2 + -35n = 0 Reorder the terms: -35n + 24n2 + 56n3 = 0 Solving -35n + 24n2 + 56n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n'. n(-35 + 24n + 56n2) = 0Subproblem 1
Set the factor 'n' equal to zero and attempt to solve: Simplifying n = 0 Solving n = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n = 0Subproblem 2
Set the factor '(-35 + 24n + 56n2)' equal to zero and attempt to solve: Simplifying -35 + 24n + 56n2 = 0 Solving -35 + 24n + 56n2 = 0 Begin completing the square. Divide all terms by 56 the coefficient of the squared term: Divide each side by '56'. -0.625 + 0.4285714286n + n2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + 0.4285714286n + 0.625 + n2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + 0.4285714286n + n2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + 0.4285714286n + n2 = 0 + 0.625 0.4285714286n + n2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 0.4285714286n + n2 = 0.625 The n term is 0.4285714286n. Take half its coefficient (0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. 0.4285714286n + 0.04591836735 + n2 = 0.625 + 0.04591836735 Reorder the terms: 0.04591836735 + 0.4285714286n + n2 = 0.625 + 0.04591836735 Combine like terms: 0.625 + 0.04591836735 = 0.67091836735 0.04591836735 + 0.4285714286n + n2 = 0.67091836735 Factor a perfect square on the left side: (n + 0.2142857143)(n + 0.2142857143) = 0.67091836735 Calculate the square root of the right side: 0.819096067 Break this problem into two subproblems by setting (n + 0.2142857143) equal to 0.819096067 and -0.819096067.Subproblem 1
n + 0.2142857143 = 0.819096067 Simplifying n + 0.2142857143 = 0.819096067 Reorder the terms: 0.2142857143 + n = 0.819096067 Solving 0.2142857143 + n = 0.819096067 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + n = 0.819096067 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + n = 0.819096067 + -0.2142857143 n = 0.819096067 + -0.2142857143 Combine like terms: 0.819096067 + -0.2142857143 = 0.6048103527 n = 0.6048103527 Simplifying n = 0.6048103527Subproblem 2
n + 0.2142857143 = -0.819096067 Simplifying n + 0.2142857143 = -0.819096067 Reorder the terms: 0.2142857143 + n = -0.819096067 Solving 0.2142857143 + n = -0.819096067 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + n = -0.819096067 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + n = -0.819096067 + -0.2142857143 n = -0.819096067 + -0.2142857143 Combine like terms: -0.819096067 + -0.2142857143 = -1.0333817813 n = -1.0333817813 Simplifying n = -1.0333817813Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.6048103527, -1.0333817813}Solution
n = {0, 0.6048103527, -1.0333817813}
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