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Simplifying 12n2 + -9n + -18 = 0 Reorder the terms: -18 + -9n + 12n2 = 0 Solving -18 + -9n + 12n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3'. 3(-6 + -3n + 4n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-6 + -3n + 4n2)' equal to zero and attempt to solve: Simplifying -6 + -3n + 4n2 = 0 Solving -6 + -3n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.5 + -0.75n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -0.75n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -0.75n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -0.75n + n2 = 0 + 1.5 -0.75n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -0.75n + n2 = 1.5 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 = 1.5 + 0.140625 Reorder the terms: 0.140625 + -0.75n + n2 = 1.5 + 0.140625 Combine like terms: 1.5 + 0.140625 = 1.640625 0.140625 + -0.75n + n2 = 1.640625 Factor a perfect square on the left side: (n + -0.375)(n + -0.375) = 1.640625 Calculate the square root of the right side: 1.280868846 Break this problem into two subproblems by setting (n + -0.375) equal to 1.280868846 and -1.280868846.Subproblem 1
n + -0.375 = 1.280868846 Simplifying n + -0.375 = 1.280868846 Reorder the terms: -0.375 + n = 1.280868846 Solving -0.375 + n = 1.280868846 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 = 1.280868846 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = 1.280868846 + 0.375 n = 1.280868846 + 0.375 Combine like terms: 1.280868846 + 0.375 = 1.655868846 n = 1.655868846 Simplifying n = 1.655868846Subproblem 2
n + -0.375 = -1.280868846 Simplifying n + -0.375 = -1.280868846 Reorder the terms: -0.375 + n = -1.280868846 Solving -0.375 + n = -1.280868846 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 = -1.280868846 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + n = -1.280868846 + 0.375 n = -1.280868846 + 0.375 Combine like terms: -1.280868846 + 0.375 = -0.905868846 n = -0.905868846 Simplifying n = -0.905868846Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.655868846, -0.905868846}Solution
n = {1.655868846, -0.905868846}
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