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Simplifying 8n2 + -16n + -6 = 0 Reorder the terms: -6 + -16n + 8n2 = 0 Solving -6 + -16n + 8n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -8n + 4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -8n + 4n2)' equal to zero and attempt to solve: Simplifying -3 + -8n + 4n2 = 0 Solving -3 + -8n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + -2n + n2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -2n + 0.75 + n2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -2n + n2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -2n + n2 = 0 + 0.75 -2n + n2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -2n + n2 = 0.75 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 0.75 + 1 Reorder the terms: 1 + -2n + n2 = 0.75 + 1 Combine like terms: 0.75 + 1 = 1.75 1 + -2n + n2 = 1.75 Factor a perfect square on the left side: (n + -1)(n + -1) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (n + -1) equal to 1.322875656 and -1.322875656.Subproblem 1
n + -1 = 1.322875656 Simplifying n + -1 = 1.322875656 Reorder the terms: -1 + n = 1.322875656 Solving -1 + n = 1.322875656 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.322875656 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.322875656 + 1 n = 1.322875656 + 1 Combine like terms: 1.322875656 + 1 = 2.322875656 n = 2.322875656 Simplifying n = 2.322875656Subproblem 2
n + -1 = -1.322875656 Simplifying n + -1 = -1.322875656 Reorder the terms: -1 + n = -1.322875656 Solving -1 + n = -1.322875656 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.322875656 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.322875656 + 1 n = -1.322875656 + 1 Combine like terms: -1.322875656 + 1 = -0.322875656 n = -0.322875656 Simplifying n = -0.322875656Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.322875656, -0.322875656}Solution
n = {2.322875656, -0.322875656}
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