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Simplifying 2n2 + 18n + -42 = 0 Reorder the terms: -42 + 18n + 2n2 = 0 Solving -42 + 18n + 2n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-21 + 9n + n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-21 + 9n + n2)' equal to zero and attempt to solve: Simplifying -21 + 9n + n2 = 0 Solving -21 + 9n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '21' to each side of the equation. -21 + 9n + 21 + n2 = 0 + 21 Reorder the terms: -21 + 21 + 9n + n2 = 0 + 21 Combine like terms: -21 + 21 = 0 0 + 9n + n2 = 0 + 21 9n + n2 = 0 + 21 Combine like terms: 0 + 21 = 21 9n + n2 = 21 The n term is 9n. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9n + 20.25 + n2 = 21 + 20.25 Reorder the terms: 20.25 + 9n + n2 = 21 + 20.25 Combine like terms: 21 + 20.25 = 41.25 20.25 + 9n + n2 = 41.25 Factor a perfect square on the left side: (n + 4.5)(n + 4.5) = 41.25 Calculate the square root of the right side: 6.422616289 Break this problem into two subproblems by setting (n + 4.5) equal to 6.422616289 and -6.422616289.Subproblem 1
n + 4.5 = 6.422616289 Simplifying n + 4.5 = 6.422616289 Reorder the terms: 4.5 + n = 6.422616289 Solving 4.5 + n = 6.422616289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 6.422616289 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 6.422616289 + -4.5 n = 6.422616289 + -4.5 Combine like terms: 6.422616289 + -4.5 = 1.922616289 n = 1.922616289 Simplifying n = 1.922616289Subproblem 2
n + 4.5 = -6.422616289 Simplifying n + 4.5 = -6.422616289 Reorder the terms: 4.5 + n = -6.422616289 Solving 4.5 + n = -6.422616289 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -6.422616289 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -6.422616289 + -4.5 n = -6.422616289 + -4.5 Combine like terms: -6.422616289 + -4.5 = -10.922616289 n = -10.922616289 Simplifying n = -10.922616289Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.922616289, -10.922616289}Solution
n = {1.922616289, -10.922616289}
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