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Simplifying 8n2 + 14n + -6 = 0 Reorder the terms: -6 + 14n + 8n2 = 0 Solving -6 + 14n + 8n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + 7n + 4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + 7n + 4n2)' equal to zero and attempt to solve: Simplifying -3 + 7n + 4n2 = 0 Solving -3 + 7n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 1.75n + n2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.75n + 0.75 + n2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.75n + n2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.75n + n2 = 0 + 0.75 1.75n + n2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.75n + n2 = 0.75 The n term is 1.75n. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75n + 0.765625 + n2 = 0.75 + 0.765625 Reorder the terms: 0.765625 + 1.75n + n2 = 0.75 + 0.765625 Combine like terms: 0.75 + 0.765625 = 1.515625 0.765625 + 1.75n + n2 = 1.515625 Factor a perfect square on the left side: (n + 0.875)(n + 0.875) = 1.515625 Calculate the square root of the right side: 1.231107225 Break this problem into two subproblems by setting (n + 0.875) equal to 1.231107225 and -1.231107225.Subproblem 1
n + 0.875 = 1.231107225 Simplifying n + 0.875 = 1.231107225 Reorder the terms: 0.875 + n = 1.231107225 Solving 0.875 + n = 1.231107225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = 1.231107225 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 1.231107225 + -0.875 n = 1.231107225 + -0.875 Combine like terms: 1.231107225 + -0.875 = 0.356107225 n = 0.356107225 Simplifying n = 0.356107225Subproblem 2
n + 0.875 = -1.231107225 Simplifying n + 0.875 = -1.231107225 Reorder the terms: 0.875 + n = -1.231107225 Solving 0.875 + n = -1.231107225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = -1.231107225 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -1.231107225 + -0.875 n = -1.231107225 + -0.875 Combine like terms: -1.231107225 + -0.875 = -2.106107225 n = -2.106107225 Simplifying n = -2.106107225Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.356107225, -2.106107225}Solution
n = {0.356107225, -2.106107225}
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