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Simplifying 12n2 + -9n = 9 Reorder the terms: -9n + 12n2 = 9 Solving -9n + 12n2 = 9 Solving for variable 'n'. Reorder the terms: -9 + -9n + 12n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + -9n + 12n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-3 + -3n + 4n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-3 + -3n + 4n2)' equal to zero and attempt to solve: Simplifying -3 + -3n + 4n2 = 0 Solving -3 + -3n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -0.75n + n2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -0.75n + 0.75 + n2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -0.75n + n2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -0.75n + n2 = 0 + 0.75 -0.75n + n2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -0.75n + n2 = 0.75 The n term is -0.75n. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75n + 0.140625 + n2 = 0.75 + 0.140625 Reorder the terms: 0.140625 + -0.75n + n2 = 0.75 + 0.140625 Combine like terms: 0.75 + 0.140625 = 0.890625 0.140625 + -0.75n + n2 = 0.890625 Factor a perfect square on the left side: (n + -0.375)(n + -0.375) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (n + -0.375) equal to 0.943729304 and -0.943729304.Subproblem 1
n + -0.375 = 0.943729304 Simplifying n + -0.375 = 0.943729304 Reorder the terms: -0.375 + n = 0.943729304 Solving -0.375 + n = 0.943729304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = 0.943729304 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 0.943729304 + 0.375 n = 0.943729304 + 0.375 Combine like terms: 0.943729304 + 0.375 = 1.318729304 n = 1.318729304 Simplifying n = 1.318729304Subproblem 2
n + -0.375 = -0.943729304 Simplifying n + -0.375 = -0.943729304 Reorder the terms: -0.375 + n = -0.943729304 Solving -0.375 + n = -0.943729304 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + n = -0.943729304 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -0.943729304 + 0.375 n = -0.943729304 + 0.375 Combine like terms: -0.943729304 + 0.375 = -0.568729304 n = -0.568729304 Simplifying n = -0.568729304Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.318729304, -0.568729304}Solution
n = {1.318729304, -0.568729304}
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