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Simplifying 8n2 + -22 = -4n Reorder the terms: -22 + 8n2 = -4n Solving -22 + 8n2 = -4n Solving for variable 'n'. Reorder the terms: -22 + 4n + 8n2 = -4n + 4n Combine like terms: -4n + 4n = 0 -22 + 4n + 8n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-11 + 2n + 4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-11 + 2n + 4n2)' equal to zero and attempt to solve: Simplifying -11 + 2n + 4n2 = 0 Solving -11 + 2n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.75 + 0.5n + n2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + 0.5n + 2.75 + n2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + 0.5n + n2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + 0.5n + n2 = 0 + 2.75 0.5n + n2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 0.5n + n2 = 2.75 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 = 2.75 + 0.0625 Reorder the terms: 0.0625 + 0.5n + n2 = 2.75 + 0.0625 Combine like terms: 2.75 + 0.0625 = 2.8125 0.0625 + 0.5n + n2 = 2.8125 Factor a perfect square on the left side: (n + 0.25)(n + 0.25) = 2.8125 Calculate the square root of the right side: 1.677050983 Break this problem into two subproblems by setting (n + 0.25) equal to 1.677050983 and -1.677050983.Subproblem 1
n + 0.25 = 1.677050983 Simplifying n + 0.25 = 1.677050983 Reorder the terms: 0.25 + n = 1.677050983 Solving 0.25 + n = 1.677050983 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 = 1.677050983 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = 1.677050983 + -0.25 n = 1.677050983 + -0.25 Combine like terms: 1.677050983 + -0.25 = 1.427050983 n = 1.427050983 Simplifying n = 1.427050983Subproblem 2
n + 0.25 = -1.677050983 Simplifying n + 0.25 = -1.677050983 Reorder the terms: 0.25 + n = -1.677050983 Solving 0.25 + n = -1.677050983 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 = -1.677050983 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + n = -1.677050983 + -0.25 n = -1.677050983 + -0.25 Combine like terms: -1.677050983 + -0.25 = -1.927050983 n = -1.927050983 Simplifying n = -1.927050983Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.427050983, -1.927050983}Solution
n = {1.427050983, -1.927050983}
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