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Simplifying (3n + 6)(6n + -3) = 9 Reorder the terms: (6 + 3n)(6n + -3) = 9 Reorder the terms: (6 + 3n)(-3 + 6n) = 9 Multiply (6 + 3n) * (-3 + 6n) (6(-3 + 6n) + 3n * (-3 + 6n)) = 9 ((-3 * 6 + 6n * 6) + 3n * (-3 + 6n)) = 9 ((-18 + 36n) + 3n * (-3 + 6n)) = 9 (-18 + 36n + (-3 * 3n + 6n * 3n)) = 9 (-18 + 36n + (-9n + 18n2)) = 9 Combine like terms: 36n + -9n = 27n (-18 + 27n + 18n2) = 9 Solving -18 + 27n + 18n2 = 9 Solving for variable 'n'. Reorder the terms: -18 + -9 + 27n + 18n2 = 9 + -9 Combine like terms: -18 + -9 = -27 -27 + 27n + 18n2 = 9 + -9 Combine like terms: 9 + -9 = 0 -27 + 27n + 18n2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(-3 + 3n + 2n2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-3 + 3n + 2n2)' equal to zero and attempt to solve: Simplifying -3 + 3n + 2n2 = 0 Solving -3 + 3n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 1.5n + n2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.5n + 1.5 + n2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.5n + n2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.5n + n2 = 0 + 1.5 1.5n + n2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.5n + n2 = 1.5 The n term is 1.5n. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5n + 0.5625 + n2 = 1.5 + 0.5625 Reorder the terms: 0.5625 + 1.5n + n2 = 1.5 + 0.5625 Combine like terms: 1.5 + 0.5625 = 2.0625 0.5625 + 1.5n + n2 = 2.0625 Factor a perfect square on the left side: (n + 0.75)(n + 0.75) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (n + 0.75) equal to 1.436140662 and -1.436140662.Subproblem 1
n + 0.75 = 1.436140662 Simplifying n + 0.75 = 1.436140662 Reorder the terms: 0.75 + n = 1.436140662 Solving 0.75 + n = 1.436140662 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = 1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = 1.436140662 + -0.75 n = 1.436140662 + -0.75 Combine like terms: 1.436140662 + -0.75 = 0.686140662 n = 0.686140662 Simplifying n = 0.686140662Subproblem 2
n + 0.75 = -1.436140662 Simplifying n + 0.75 = -1.436140662 Reorder the terms: 0.75 + n = -1.436140662 Solving 0.75 + n = -1.436140662 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + n = -1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + n = -1.436140662 + -0.75 n = -1.436140662 + -0.75 Combine like terms: -1.436140662 + -0.75 = -2.186140662 n = -2.186140662 Simplifying n = -2.186140662Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.686140662, -2.186140662}Solution
n = {0.686140662, -2.186140662}
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