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Simplifying (6n2 + -2n3 + -2n) + (7n2 + 8n + -2n3) = 0 Reorder the terms: (-2n + 6n2 + -2n3) + (7n2 + 8n + -2n3) = 0 Remove parenthesis around (-2n + 6n2 + -2n3) -2n + 6n2 + -2n3 + (7n2 + 8n + -2n3) = 0 Reorder the terms: -2n + 6n2 + -2n3 + (8n + 7n2 + -2n3) = 0 Remove parenthesis around (8n + 7n2 + -2n3) -2n + 6n2 + -2n3 + 8n + 7n2 + -2n3 = 0 Reorder the terms: -2n + 8n + 6n2 + 7n2 + -2n3 + -2n3 = 0 Combine like terms: -2n + 8n = 6n 6n + 6n2 + 7n2 + -2n3 + -2n3 = 0 Combine like terms: 6n2 + 7n2 = 13n2 6n + 13n2 + -2n3 + -2n3 = 0 Combine like terms: -2n3 + -2n3 = -4n3 6n + 13n2 + -4n3 = 0 Solving 6n + 13n2 + -4n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), 'n'. n(6 + 13n + -4n2) = 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 '(6 + 13n + -4n2)' equal to zero and attempt to solve: Simplifying 6 + 13n + -4n2 = 0 Solving 6 + 13n + -4n2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -1.5 + -3.25n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -3.25n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -3.25n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -3.25n + n2 = 0 + 1.5 -3.25n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -3.25n + n2 = 1.5 The n term is -3.25n. Take half its coefficient (-1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. -3.25n + 2.640625 + n2 = 1.5 + 2.640625 Reorder the terms: 2.640625 + -3.25n + n2 = 1.5 + 2.640625 Combine like terms: 1.5 + 2.640625 = 4.140625 2.640625 + -3.25n + n2 = 4.140625 Factor a perfect square on the left side: (n + -1.625)(n + -1.625) = 4.140625 Calculate the square root of the right side: 2.034852575 Break this problem into two subproblems by setting (n + -1.625) equal to 2.034852575 and -2.034852575.Subproblem 1
n + -1.625 = 2.034852575 Simplifying n + -1.625 = 2.034852575 Reorder the terms: -1.625 + n = 2.034852575 Solving -1.625 + n = 2.034852575 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + n = 2.034852575 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + n = 2.034852575 + 1.625 n = 2.034852575 + 1.625 Combine like terms: 2.034852575 + 1.625 = 3.659852575 n = 3.659852575 Simplifying n = 3.659852575Subproblem 2
n + -1.625 = -2.034852575 Simplifying n + -1.625 = -2.034852575 Reorder the terms: -1.625 + n = -2.034852575 Solving -1.625 + n = -2.034852575 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.625' to each side of the equation. -1.625 + 1.625 + n = -2.034852575 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + n = -2.034852575 + 1.625 n = -2.034852575 + 1.625 Combine like terms: -2.034852575 + 1.625 = -0.409852575 n = -0.409852575 Simplifying n = -0.409852575Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.659852575, -0.409852575}Solution
n = {0, 3.659852575, -0.409852575}
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