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Simplifying 8n2 + -42n + -13 = 3 Reorder the terms: -13 + -42n + 8n2 = 3 Solving -13 + -42n + 8n2 = 3 Solving for variable 'n'. Reorder the terms: -13 + -3 + -42n + 8n2 = 3 + -3 Combine like terms: -13 + -3 = -16 -16 + -42n + 8n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -16 + -42n + 8n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-8 + -21n + 4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-8 + -21n + 4n2)' equal to zero and attempt to solve: Simplifying -8 + -21n + 4n2 = 0 Solving -8 + -21n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2 + -5.25n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -5.25n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + -5.25n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -5.25n + n2 = 0 + 2 -5.25n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 -5.25n + n2 = 2 The n term is -5.25n. Take half its coefficient (-2.625). Square it (6.890625) and add it to both sides. Add '6.890625' to each side of the equation. -5.25n + 6.890625 + n2 = 2 + 6.890625 Reorder the terms: 6.890625 + -5.25n + n2 = 2 + 6.890625 Combine like terms: 2 + 6.890625 = 8.890625 6.890625 + -5.25n + n2 = 8.890625 Factor a perfect square on the left side: (n + -2.625)(n + -2.625) = 8.890625 Calculate the square root of the right side: 2.98171511 Break this problem into two subproblems by setting (n + -2.625) equal to 2.98171511 and -2.98171511.Subproblem 1
n + -2.625 = 2.98171511 Simplifying n + -2.625 = 2.98171511 Reorder the terms: -2.625 + n = 2.98171511 Solving -2.625 + n = 2.98171511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + n = 2.98171511 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + n = 2.98171511 + 2.625 n = 2.98171511 + 2.625 Combine like terms: 2.98171511 + 2.625 = 5.60671511 n = 5.60671511 Simplifying n = 5.60671511Subproblem 2
n + -2.625 = -2.98171511 Simplifying n + -2.625 = -2.98171511 Reorder the terms: -2.625 + n = -2.98171511 Solving -2.625 + n = -2.98171511 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + n = -2.98171511 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + n = -2.98171511 + 2.625 n = -2.98171511 + 2.625 Combine like terms: -2.98171511 + 2.625 = -0.35671511 n = -0.35671511 Simplifying n = -0.35671511Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.60671511, -0.35671511}Solution
n = {5.60671511, -0.35671511}
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