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Simplifying 8n2 + 52n = 6438 Reorder the terms: 52n + 8n2 = 6438 Solving 52n + 8n2 = 6438 Solving for variable 'n'. Reorder the terms: -6438 + 52n + 8n2 = 6438 + -6438 Combine like terms: 6438 + -6438 = 0 -6438 + 52n + 8n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3219 + 26n + 4n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3219 + 26n + 4n2)' equal to zero and attempt to solve: Simplifying -3219 + 26n + 4n2 = 0 Solving -3219 + 26n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -804.75 + 6.5n + n2 = 0 Move the constant term to the right: Add '804.75' to each side of the equation. -804.75 + 6.5n + 804.75 + n2 = 0 + 804.75 Reorder the terms: -804.75 + 804.75 + 6.5n + n2 = 0 + 804.75 Combine like terms: -804.75 + 804.75 = 0.00 0.00 + 6.5n + n2 = 0 + 804.75 6.5n + n2 = 0 + 804.75 Combine like terms: 0 + 804.75 = 804.75 6.5n + n2 = 804.75 The n term is 6.5n. Take half its coefficient (3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. 6.5n + 10.5625 + n2 = 804.75 + 10.5625 Reorder the terms: 10.5625 + 6.5n + n2 = 804.75 + 10.5625 Combine like terms: 804.75 + 10.5625 = 815.3125 10.5625 + 6.5n + n2 = 815.3125 Factor a perfect square on the left side: (n + 3.25)(n + 3.25) = 815.3125 Calculate the square root of the right side: 28.553677521 Break this problem into two subproblems by setting (n + 3.25) equal to 28.553677521 and -28.553677521.Subproblem 1
n + 3.25 = 28.553677521 Simplifying n + 3.25 = 28.553677521 Reorder the terms: 3.25 + n = 28.553677521 Solving 3.25 + n = 28.553677521 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + n = 28.553677521 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + n = 28.553677521 + -3.25 n = 28.553677521 + -3.25 Combine like terms: 28.553677521 + -3.25 = 25.303677521 n = 25.303677521 Simplifying n = 25.303677521Subproblem 2
n + 3.25 = -28.553677521 Simplifying n + 3.25 = -28.553677521 Reorder the terms: 3.25 + n = -28.553677521 Solving 3.25 + n = -28.553677521 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + n = -28.553677521 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + n = -28.553677521 + -3.25 n = -28.553677521 + -3.25 Combine like terms: -28.553677521 + -3.25 = -31.803677521 n = -31.803677521 Simplifying n = -31.803677521Solution
The solution to the problem is based on the solutions from the subproblems. n = {25.303677521, -31.803677521}Solution
n = {25.303677521, -31.803677521}
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