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Simplifying 3n2 + -15n = 3 Reorder the terms: -15n + 3n2 = 3 Solving -15n + 3n2 = 3 Solving for variable 'n'. Reorder the terms: -3 + -15n + 3n2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + -15n + 3n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + -5n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + -5n + n2)' equal to zero and attempt to solve: Simplifying -1 + -5n + n2 = 0 Solving -1 + -5n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -5n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + -5n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -5n + n2 = 0 + 1 -5n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 -5n + n2 = 1 The n term is -5n. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5n + 6.25 + n2 = 1 + 6.25 Reorder the terms: 6.25 + -5n + n2 = 1 + 6.25 Combine like terms: 1 + 6.25 = 7.25 6.25 + -5n + n2 = 7.25 Factor a perfect square on the left side: (n + -2.5)(n + -2.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (n + -2.5) equal to 2.692582404 and -2.692582404.Subproblem 1
n + -2.5 = 2.692582404 Simplifying n + -2.5 = 2.692582404 Reorder the terms: -2.5 + n = 2.692582404 Solving -2.5 + n = 2.692582404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = 2.692582404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = 2.692582404 + 2.5 n = 2.692582404 + 2.5 Combine like terms: 2.692582404 + 2.5 = 5.192582404 n = 5.192582404 Simplifying n = 5.192582404Subproblem 2
n + -2.5 = -2.692582404 Simplifying n + -2.5 = -2.692582404 Reorder the terms: -2.5 + n = -2.692582404 Solving -2.5 + n = -2.692582404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + n = -2.692582404 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + n = -2.692582404 + 2.5 n = -2.692582404 + 2.5 Combine like terms: -2.692582404 + 2.5 = -0.192582404 n = -0.192582404 Simplifying n = -0.192582404Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.192582404, -0.192582404}Solution
n = {5.192582404, -0.192582404}
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