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Simplifying 2(3n + -4) * 7n + -18 = 0 Reorder the terms: 2(-4 + 3n) * 7n + -18 = 0 Reorder the terms for easier multiplication: 2 * 7n(-4 + 3n) + -18 = 0 Multiply 2 * 7 14n(-4 + 3n) + -18 = 0 (-4 * 14n + 3n * 14n) + -18 = 0 (-56n + 42n2) + -18 = 0 Reorder the terms: -18 + -56n + 42n2 = 0 Solving -18 + -56n + 42n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-9 + -28n + 21n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-9 + -28n + 21n2)' equal to zero and attempt to solve: Simplifying -9 + -28n + 21n2 = 0 Solving -9 + -28n + 21n2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.4285714286 + -1.333333333n + n2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + -1.333333333n + 0.4285714286 + n2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + -1.333333333n + n2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + -1.333333333n + n2 = 0 + 0.4285714286 -1.333333333n + n2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 -1.333333333n + n2 = 0.4285714286 The n term is -1.333333333n. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333n + 0.4444444442 + n2 = 0.4285714286 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333n + n2 = 0.4285714286 + 0.4444444442 Combine like terms: 0.4285714286 + 0.4444444442 = 0.8730158728 0.4444444442 + -1.333333333n + n2 = 0.8730158728 Factor a perfect square on the left side: (n + -0.6666666665)(n + -0.6666666665) = 0.8730158728 Calculate the square root of the right side: 0.934353184 Break this problem into two subproblems by setting (n + -0.6666666665) equal to 0.934353184 and -0.934353184.Subproblem 1
n + -0.6666666665 = 0.934353184 Simplifying n + -0.6666666665 = 0.934353184 Reorder the terms: -0.6666666665 + n = 0.934353184 Solving -0.6666666665 + n = 0.934353184 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + n = 0.934353184 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + n = 0.934353184 + 0.6666666665 n = 0.934353184 + 0.6666666665 Combine like terms: 0.934353184 + 0.6666666665 = 1.6010198505 n = 1.6010198505 Simplifying n = 1.6010198505Subproblem 2
n + -0.6666666665 = -0.934353184 Simplifying n + -0.6666666665 = -0.934353184 Reorder the terms: -0.6666666665 + n = -0.934353184 Solving -0.6666666665 + n = -0.934353184 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + n = -0.934353184 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + n = -0.934353184 + 0.6666666665 n = -0.934353184 + 0.6666666665 Combine like terms: -0.934353184 + 0.6666666665 = -0.2676865175 n = -0.2676865175 Simplifying n = -0.2676865175Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.6010198505, -0.2676865175}Solution
n = {1.6010198505, -0.2676865175}
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