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Simplifying 8n2 + -4n = 18 Reorder the terms: -4n + 8n2 = 18 Solving -4n + 8n2 = 18 Solving for variable 'n'. Reorder the terms: -18 + -4n + 8n2 = 18 + -18 Combine like terms: 18 + -18 = 0 -18 + -4n + 8n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + -2n + 4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + -2n + 4n2)' equal to zero and attempt to solve: Simplifying -9 + -2n + 4n2 = 0 Solving -9 + -2n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.25 + -0.5n + n2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + -0.5n + 2.25 + n2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + -0.5n + n2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + -0.5n + n2 = 0 + 2.25 -0.5n + n2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 -0.5n + n2 = 2.25 The n term is -0.5n. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5n + 0.0625 + n2 = 2.25 + 0.0625 Reorder the terms: 0.0625 + -0.5n + n2 = 2.25 + 0.0625 Combine like terms: 2.25 + 0.0625 = 2.3125 0.0625 + -0.5n + n2 = 2.3125 Factor a perfect square on the left side: (n + -0.25)(n + -0.25) = 2.3125 Calculate the square root of the right side: 1.520690633 Break this problem into two subproblems by setting (n + -0.25) equal to 1.520690633 and -1.520690633.Subproblem 1
n + -0.25 = 1.520690633 Simplifying n + -0.25 = 1.520690633 Reorder the terms: -0.25 + n = 1.520690633 Solving -0.25 + n = 1.520690633 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = 1.520690633 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = 1.520690633 + 0.25 n = 1.520690633 + 0.25 Combine like terms: 1.520690633 + 0.25 = 1.770690633 n = 1.770690633 Simplifying n = 1.770690633Subproblem 2
n + -0.25 = -1.520690633 Simplifying n + -0.25 = -1.520690633 Reorder the terms: -0.25 + n = -1.520690633 Solving -0.25 + n = -1.520690633 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + n = -1.520690633 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + n = -1.520690633 + 0.25 n = -1.520690633 + 0.25 Combine like terms: -1.520690633 + 0.25 = -1.270690633 n = -1.270690633 Simplifying n = -1.270690633Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.770690633, -1.270690633}Solution
n = {1.770690633, -1.270690633}
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