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Simplifying 8n(2n + -3) = 12 Reorder the terms: 8n(-3 + 2n) = 12 (-3 * 8n + 2n * 8n) = 12 (-24n + 16n2) = 12 Solving -24n + 16n2 = 12 Solving for variable 'n'. Reorder the terms: -12 + -24n + 16n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + -24n + 16n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -6n + 4n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -6n + 4n2)' equal to zero and attempt to solve: Simplifying -3 + -6n + 4n2 = 0 Solving -3 + -6n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -1.5n + n2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -1.5n + 0.75 + n2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -1.5n + n2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -1.5n + n2 = 0 + 0.75 -1.5n + n2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -1.5n + n2 = 0.75 The n term is -1.5n. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5n + 0.5625 + n2 = 0.75 + 0.5625 Reorder the terms: 0.5625 + -1.5n + n2 = 0.75 + 0.5625 Combine like terms: 0.75 + 0.5625 = 1.3125 0.5625 + -1.5n + n2 = 1.3125 Factor a perfect square on the left side: (n + -0.75)(n + -0.75) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (n + -0.75) equal to 1.145643924 and -1.145643924.Subproblem 1
n + -0.75 = 1.145643924 Simplifying n + -0.75 = 1.145643924 Reorder the terms: -0.75 + n = 1.145643924 Solving -0.75 + n = 1.145643924 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 1.145643924 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 1.145643924 + 0.75 n = 1.145643924 + 0.75 Combine like terms: 1.145643924 + 0.75 = 1.895643924 n = 1.895643924 Simplifying n = 1.895643924Subproblem 2
n + -0.75 = -1.145643924 Simplifying n + -0.75 = -1.145643924 Reorder the terms: -0.75 + n = -1.145643924 Solving -0.75 + n = -1.145643924 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -1.145643924 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -1.145643924 + 0.75 n = -1.145643924 + 0.75 Combine like terms: -1.145643924 + 0.75 = -0.395643924 n = -0.395643924 Simplifying n = -0.395643924Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.895643924, -0.395643924}Solution
n = {1.895643924, -0.395643924}
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