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Simplifying 3n2 + 21n + -288 = 0 Reorder the terms: -288 + 21n + 3n2 = 0 Solving -288 + 21n + 3n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '3'. 3(-96 + 7n + n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-96 + 7n + n2)' equal to zero and attempt to solve: Simplifying -96 + 7n + n2 = 0 Solving -96 + 7n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '96' to each side of the equation. -96 + 7n + 96 + n2 = 0 + 96 Reorder the terms: -96 + 96 + 7n + n2 = 0 + 96 Combine like terms: -96 + 96 = 0 0 + 7n + n2 = 0 + 96 7n + n2 = 0 + 96 Combine like terms: 0 + 96 = 96 7n + n2 = 96 The n term is 7n. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7n + 12.25 + n2 = 96 + 12.25 Reorder the terms: 12.25 + 7n + n2 = 96 + 12.25 Combine like terms: 96 + 12.25 = 108.25 12.25 + 7n + n2 = 108.25 Factor a perfect square on the left side: (n + 3.5)(n + 3.5) = 108.25 Calculate the square root of the right side: 10.404326023 Break this problem into two subproblems by setting (n + 3.5) equal to 10.404326023 and -10.404326023.Subproblem 1
n + 3.5 = 10.404326023 Simplifying n + 3.5 = 10.404326023 Reorder the terms: 3.5 + n = 10.404326023 Solving 3.5 + n = 10.404326023 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = 10.404326023 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = 10.404326023 + -3.5 n = 10.404326023 + -3.5 Combine like terms: 10.404326023 + -3.5 = 6.904326023 n = 6.904326023 Simplifying n = 6.904326023Subproblem 2
n + 3.5 = -10.404326023 Simplifying n + 3.5 = -10.404326023 Reorder the terms: 3.5 + n = -10.404326023 Solving 3.5 + n = -10.404326023 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + n = -10.404326023 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + n = -10.404326023 + -3.5 n = -10.404326023 + -3.5 Combine like terms: -10.404326023 + -3.5 = -13.904326023 n = -13.904326023 Simplifying n = -13.904326023Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.904326023, -13.904326023}Solution
n = {6.904326023, -13.904326023}
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