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Simplifying -4n(n + -6) = 12 Reorder the terms: -4n(-6 + n) = 12 (-6 * -4n + n * -4n) = 12 (24n + -4n2) = 12 Solving 24n + -4n2 = 12 Solving for variable 'n'. Reorder the terms: -12 + 24n + -4n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + 24n + -4n2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + 6n + -1n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + 6n + -1n2)' equal to zero and attempt to solve: Simplifying -3 + 6n + -1n2 = 0 Solving -3 + 6n + -1n2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 3 + -6n + n2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -6n + -3 + n2 = 0 + -3 Reorder the terms: 3 + -3 + -6n + n2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -6n + n2 = 0 + -3 -6n + n2 = 0 + -3 Combine like terms: 0 + -3 = -3 -6n + n2 = -3 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = -3 + 9 Reorder the terms: 9 + -6n + n2 = -3 + 9 Combine like terms: -3 + 9 = 6 9 + -6n + n2 = 6 Factor a perfect square on the left side: (n + -3)(n + -3) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (n + -3) equal to 2.449489743 and -2.449489743.Subproblem 1
n + -3 = 2.449489743 Simplifying n + -3 = 2.449489743 Reorder the terms: -3 + n = 2.449489743 Solving -3 + n = 2.449489743 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.449489743 + 3 n = 2.449489743 + 3 Combine like terms: 2.449489743 + 3 = 5.449489743 n = 5.449489743 Simplifying n = 5.449489743Subproblem 2
n + -3 = -2.449489743 Simplifying n + -3 = -2.449489743 Reorder the terms: -3 + n = -2.449489743 Solving -3 + n = -2.449489743 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -2.449489743 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.449489743 + 3 n = -2.449489743 + 3 Combine like terms: -2.449489743 + 3 = 0.550510257 n = 0.550510257 Simplifying n = 0.550510257Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.449489743, 0.550510257}Solution
n = {5.449489743, 0.550510257}
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