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Simplifying -3n(n + -6) + -3 = -3n + 9 Reorder the terms: -3n(-6 + n) + -3 = -3n + 9 (-6 * -3n + n * -3n) + -3 = -3n + 9 (18n + -3n2) + -3 = -3n + 9 Reorder the terms: -3 + 18n + -3n2 = -3n + 9 Reorder the terms: -3 + 18n + -3n2 = 9 + -3n Solving -3 + 18n + -3n2 = 9 + -3n Solving for variable 'n'. Reorder the terms: -3 + -9 + 18n + 3n + -3n2 = 9 + -3n + -9 + 3n Combine like terms: -3 + -9 = -12 -12 + 18n + 3n + -3n2 = 9 + -3n + -9 + 3n Combine like terms: 18n + 3n = 21n -12 + 21n + -3n2 = 9 + -3n + -9 + 3n Reorder the terms: -12 + 21n + -3n2 = 9 + -9 + -3n + 3n Combine like terms: 9 + -9 = 0 -12 + 21n + -3n2 = 0 + -3n + 3n -12 + 21n + -3n2 = -3n + 3n Combine like terms: -3n + 3n = 0 -12 + 21n + -3n2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-4 + 7n + -1n2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-4 + 7n + -1n2)' equal to zero and attempt to solve: Simplifying -4 + 7n + -1n2 = 0 Solving -4 + 7n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 4 + -7n + n2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -7n + -4 + n2 = 0 + -4 Reorder the terms: 4 + -4 + -7n + n2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -7n + n2 = 0 + -4 -7n + n2 = 0 + -4 Combine like terms: 0 + -4 = -4 -7n + n2 = -4 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 = -4 + 12.25 Reorder the terms: 12.25 + -7n + n2 = -4 + 12.25 Combine like terms: -4 + 12.25 = 8.25 12.25 + -7n + n2 = 8.25 Factor a perfect square on the left side: (n + -3.5)(n + -3.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (n + -3.5) equal to 2.872281323 and -2.872281323.Subproblem 1
n + -3.5 = 2.872281323 Simplifying n + -3.5 = 2.872281323 Reorder the terms: -3.5 + n = 2.872281323 Solving -3.5 + n = 2.872281323 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 = 2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = 2.872281323 + 3.5 n = 2.872281323 + 3.5 Combine like terms: 2.872281323 + 3.5 = 6.372281323 n = 6.372281323 Simplifying n = 6.372281323Subproblem 2
n + -3.5 = -2.872281323 Simplifying n + -3.5 = -2.872281323 Reorder the terms: -3.5 + n = -2.872281323 Solving -3.5 + n = -2.872281323 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 = -2.872281323 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + n = -2.872281323 + 3.5 n = -2.872281323 + 3.5 Combine like terms: -2.872281323 + 3.5 = 0.627718677 n = 0.627718677 Simplifying n = 0.627718677Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.372281323, 0.627718677}Solution
n = {6.372281323, 0.627718677}
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