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Simplifying -8n2 + -12n + 14 = 8 Reorder the terms: 14 + -12n + -8n2 = 8 Solving 14 + -12n + -8n2 = 8 Solving for variable 'n'. Reorder the terms: 14 + -8 + -12n + -8n2 = 8 + -8 Combine like terms: 14 + -8 = 6 6 + -12n + -8n2 = 8 + -8 Combine like terms: 8 + -8 = 0 6 + -12n + -8n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(3 + -6n + -4n2) = 0 Ignore the factor 2.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 = 0.395643924 n = 0.395643924 Simplifying n = 0.395643924Subproblem 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 = -1.895643924 n = -1.895643924 Simplifying n = -1.895643924Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.395643924, -1.895643924}Solution
n = {0.395643924, -1.895643924}
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