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Simplifying 10n2 = 8 + -12n Solving 10n2 = 8 + -12n Solving for variable 'n'. Reorder the terms: -8 + 12n + 10n2 = 8 + -12n + -8 + 12n Reorder the terms: -8 + 12n + 10n2 = 8 + -8 + -12n + 12n Combine like terms: 8 + -8 = 0 -8 + 12n + 10n2 = 0 + -12n + 12n -8 + 12n + 10n2 = -12n + 12n Combine like terms: -12n + 12n = 0 -8 + 12n + 10n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + 6n + 5n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + 6n + 5n2)' equal to zero and attempt to solve: Simplifying -4 + 6n + 5n2 = 0 Solving -4 + 6n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.8 + 1.2n + n2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + 1.2n + 0.8 + n2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + 1.2n + n2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + 1.2n + n2 = 0 + 0.8 1.2n + n2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 1.2n + n2 = 0.8 The n term is 1.2n. Take half its coefficient (0.6). Square it (0.36) and add it to both sides. Add '0.36' to each side of the equation. 1.2n + 0.36 + n2 = 0.8 + 0.36 Reorder the terms: 0.36 + 1.2n + n2 = 0.8 + 0.36 Combine like terms: 0.8 + 0.36 = 1.16 0.36 + 1.2n + n2 = 1.16 Factor a perfect square on the left side: (n + 0.6)(n + 0.6) = 1.16 Calculate the square root of the right side: 1.077032961 Break this problem into two subproblems by setting (n + 0.6) equal to 1.077032961 and -1.077032961.Subproblem 1
n + 0.6 = 1.077032961 Simplifying n + 0.6 = 1.077032961 Reorder the terms: 0.6 + n = 1.077032961 Solving 0.6 + n = 1.077032961 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + n = 1.077032961 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + n = 1.077032961 + -0.6 n = 1.077032961 + -0.6 Combine like terms: 1.077032961 + -0.6 = 0.477032961 n = 0.477032961 Simplifying n = 0.477032961Subproblem 2
n + 0.6 = -1.077032961 Simplifying n + 0.6 = -1.077032961 Reorder the terms: 0.6 + n = -1.077032961 Solving 0.6 + n = -1.077032961 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6' to each side of the equation. 0.6 + -0.6 + n = -1.077032961 + -0.6 Combine like terms: 0.6 + -0.6 = 0.0 0.0 + n = -1.077032961 + -0.6 n = -1.077032961 + -0.6 Combine like terms: -1.077032961 + -0.6 = -1.677032961 n = -1.677032961 Simplifying n = -1.677032961Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.477032961, -1.677032961}Solution
n = {0.477032961, -1.677032961}
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