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Simplifying 2n2 + -6n = 28 Reorder the terms: -6n + 2n2 = 28 Solving -6n + 2n2 = 28 Solving for variable 'n'. Reorder the terms: -28 + -6n + 2n2 = 28 + -28 Combine like terms: 28 + -28 = 0 -28 + -6n + 2n2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-14 + -3n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-14 + -3n + n2)' equal to zero and attempt to solve: Simplifying -14 + -3n + n2 = 0 Solving -14 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + -3n + 14 + n2 = 0 + 14 Reorder the terms: -14 + 14 + -3n + n2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + -3n + n2 = 0 + 14 -3n + n2 = 0 + 14 Combine like terms: 0 + 14 = 14 -3n + n2 = 14 The n term is -3n. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3n + 2.25 + n2 = 14 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 14 + 2.25 Combine like terms: 14 + 2.25 = 16.25 2.25 + -3n + n2 = 16.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 16.25 Calculate the square root of the right side: 4.031128874 Break this problem into two subproblems by setting (n + -1.5) equal to 4.031128874 and -4.031128874.Subproblem 1
n + -1.5 = 4.031128874 Simplifying n + -1.5 = 4.031128874 Reorder the terms: -1.5 + n = 4.031128874 Solving -1.5 + n = 4.031128874 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = 4.031128874 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 4.031128874 + 1.5 n = 4.031128874 + 1.5 Combine like terms: 4.031128874 + 1.5 = 5.531128874 n = 5.531128874 Simplifying n = 5.531128874Subproblem 2
n + -1.5 = -4.031128874 Simplifying n + -1.5 = -4.031128874 Reorder the terms: -1.5 + n = -4.031128874 Solving -1.5 + n = -4.031128874 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + n = -4.031128874 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -4.031128874 + 1.5 n = -4.031128874 + 1.5 Combine like terms: -4.031128874 + 1.5 = -2.531128874 n = -2.531128874 Simplifying n = -2.531128874Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.531128874, -2.531128874}Solution
n = {5.531128874, -2.531128874}
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