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Simplifying 4n2 + 16n + -54 = 0 Reorder the terms: -54 + 16n + 4n2 = 0 Solving -54 + 16n + 4n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-27 + 8n + 2n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-27 + 8n + 2n2)' equal to zero and attempt to solve: Simplifying -27 + 8n + 2n2 = 0 Solving -27 + 8n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -13.5 + 4n + n2 = 0 Move the constant term to the right: Add '13.5' to each side of the equation. -13.5 + 4n + 13.5 + n2 = 0 + 13.5 Reorder the terms: -13.5 + 13.5 + 4n + n2 = 0 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + 4n + n2 = 0 + 13.5 4n + n2 = 0 + 13.5 Combine like terms: 0 + 13.5 = 13.5 4n + n2 = 13.5 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = 13.5 + 4 Reorder the terms: 4 + 4n + n2 = 13.5 + 4 Combine like terms: 13.5 + 4 = 17.5 4 + 4n + n2 = 17.5 Factor a perfect square on the left side: (n + 2)(n + 2) = 17.5 Calculate the square root of the right side: 4.183300133 Break this problem into two subproblems by setting (n + 2) equal to 4.183300133 and -4.183300133.Subproblem 1
n + 2 = 4.183300133 Simplifying n + 2 = 4.183300133 Reorder the terms: 2 + n = 4.183300133 Solving 2 + n = 4.183300133 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 4.183300133 + -2 Combine like terms: 2 + -2 = 0 0 + n = 4.183300133 + -2 n = 4.183300133 + -2 Combine like terms: 4.183300133 + -2 = 2.183300133 n = 2.183300133 Simplifying n = 2.183300133Subproblem 2
n + 2 = -4.183300133 Simplifying n + 2 = -4.183300133 Reorder the terms: 2 + n = -4.183300133 Solving 2 + n = -4.183300133 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -4.183300133 + -2 Combine like terms: 2 + -2 = 0 0 + n = -4.183300133 + -2 n = -4.183300133 + -2 Combine like terms: -4.183300133 + -2 = -6.183300133 n = -6.183300133 Simplifying n = -6.183300133Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.183300133, -6.183300133}Solution
n = {2.183300133, -6.183300133}
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