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Simplifying 6n2 + -6n = 12n + 36 Reorder the terms: -6n + 6n2 = 12n + 36 Reorder the terms: -6n + 6n2 = 36 + 12n Solving -6n + 6n2 = 36 + 12n Solving for variable 'n'. Reorder the terms: -36 + -6n + -12n + 6n2 = 36 + 12n + -36 + -12n Combine like terms: -6n + -12n = -18n -36 + -18n + 6n2 = 36 + 12n + -36 + -12n Reorder the terms: -36 + -18n + 6n2 = 36 + -36 + 12n + -12n Combine like terms: 36 + -36 = 0 -36 + -18n + 6n2 = 0 + 12n + -12n -36 + -18n + 6n2 = 12n + -12n Combine like terms: 12n + -12n = 0 -36 + -18n + 6n2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-6 + -3n + n2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-6 + -3n + n2)' equal to zero and attempt to solve: Simplifying -6 + -3n + n2 = 0 Solving -6 + -3n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '6' to each side of the equation. -6 + -3n + 6 + n2 = 0 + 6 Reorder the terms: -6 + 6 + -3n + n2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + -3n + n2 = 0 + 6 -3n + n2 = 0 + 6 Combine like terms: 0 + 6 = 6 -3n + n2 = 6 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 = 6 + 2.25 Reorder the terms: 2.25 + -3n + n2 = 6 + 2.25 Combine like terms: 6 + 2.25 = 8.25 2.25 + -3n + n2 = 8.25 Factor a perfect square on the left side: (n + -1.5)(n + -1.5) = 8.25 Calculate the square root of the right side: 2.872281323 Break this problem into two subproblems by setting (n + -1.5) equal to 2.872281323 and -2.872281323.Subproblem 1
n + -1.5 = 2.872281323 Simplifying n + -1.5 = 2.872281323 Reorder the terms: -1.5 + n = 2.872281323 Solving -1.5 + n = 2.872281323 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 = 2.872281323 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = 2.872281323 + 1.5 n = 2.872281323 + 1.5 Combine like terms: 2.872281323 + 1.5 = 4.372281323 n = 4.372281323 Simplifying n = 4.372281323Subproblem 2
n + -1.5 = -2.872281323 Simplifying n + -1.5 = -2.872281323 Reorder the terms: -1.5 + n = -2.872281323 Solving -1.5 + n = -2.872281323 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 = -2.872281323 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + n = -2.872281323 + 1.5 n = -2.872281323 + 1.5 Combine like terms: -2.872281323 + 1.5 = -1.372281323 n = -1.372281323 Simplifying n = -1.372281323Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.372281323, -1.372281323}Solution
n = {4.372281323, -1.372281323}
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