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Simplifying 16n2 + 16n + -64 = 0 Reorder the terms: -64 + 16n + 16n2 = 0 Solving -64 + 16n + 16n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '16'. 16(-4 + n + n2) = 0 Ignore the factor 16.Subproblem 1
Set the factor '(-4 + n + n2)' equal to zero and attempt to solve: Simplifying -4 + n + n2 = 0 Solving -4 + n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '4' to each side of the equation. -4 + n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + n + n2 = 0 + 4 n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 n + n2 = 4 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. n + 0.25 + n2 = 4 + 0.25 Reorder the terms: 0.25 + n + n2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 0.25 + n + n2 = 4.25 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (n + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
n + 0.5 = 2.061552813 Simplifying n + 0.5 = 2.061552813 Reorder the terms: 0.5 + n = 2.061552813 Solving 0.5 + n = 2.061552813 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 2.061552813 + -0.5 n = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 n = 1.561552813 Simplifying n = 1.561552813Subproblem 2
n + 0.5 = -2.061552813 Simplifying n + 0.5 = -2.061552813 Reorder the terms: 0.5 + n = -2.061552813 Solving 0.5 + n = -2.061552813 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -2.061552813 + -0.5 n = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 n = -2.561552813 Simplifying n = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.561552813, -2.561552813}Solution
n = {1.561552813, -2.561552813}
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