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Simplifying 2n2 + 6n + -214 = 0 Reorder the terms: -214 + 6n + 2n2 = 0 Solving -214 + 6n + 2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-107 + 3n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-107 + 3n + n2)' equal to zero and attempt to solve: Simplifying -107 + 3n + n2 = 0 Solving -107 + 3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '107' to each side of the equation. -107 + 3n + 107 + n2 = 0 + 107 Reorder the terms: -107 + 107 + 3n + n2 = 0 + 107 Combine like terms: -107 + 107 = 0 0 + 3n + n2 = 0 + 107 3n + n2 = 0 + 107 Combine like terms: 0 + 107 = 107 3n + n2 = 107 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 = 107 + 2.25 Reorder the terms: 2.25 + 3n + n2 = 107 + 2.25 Combine like terms: 107 + 2.25 = 109.25 2.25 + 3n + n2 = 109.25 Factor a perfect square on the left side: (n + 1.5)(n + 1.5) = 109.25 Calculate the square root of the right side: 10.45227248 Break this problem into two subproblems by setting (n + 1.5) equal to 10.45227248 and -10.45227248.Subproblem 1
n + 1.5 = 10.45227248 Simplifying n + 1.5 = 10.45227248 Reorder the terms: 1.5 + n = 10.45227248 Solving 1.5 + n = 10.45227248 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 = 10.45227248 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = 10.45227248 + -1.5 n = 10.45227248 + -1.5 Combine like terms: 10.45227248 + -1.5 = 8.95227248 n = 8.95227248 Simplifying n = 8.95227248Subproblem 2
n + 1.5 = -10.45227248 Simplifying n + 1.5 = -10.45227248 Reorder the terms: 1.5 + n = -10.45227248 Solving 1.5 + n = -10.45227248 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 = -10.45227248 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + n = -10.45227248 + -1.5 n = -10.45227248 + -1.5 Combine like terms: -10.45227248 + -1.5 = -11.95227248 n = -11.95227248 Simplifying n = -11.95227248Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.95227248, -11.95227248}Solution
n = {8.95227248, -11.95227248}
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