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Simplifying 3n2 + 4n + -6 = 0 Reorder the terms: -6 + 4n + 3n2 = 0 Solving -6 + 4n + 3n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + 1.333333333n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.333333333n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + 1.333333333n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.333333333n + n2 = 0 + 2 1.333333333n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.333333333n + n2 = 2 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 = 2 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333n + n2 = 2 + 0.4444444442 Combine like terms: 2 + 0.4444444442 = 2.4444444442 0.4444444442 + 1.333333333n + n2 = 2.4444444442 Factor a perfect square on the left side: (n + 0.6666666665)(n + 0.6666666665) = 2.4444444442 Calculate the square root of the right side: 1.56347192 Break this problem into two subproblems by setting (n + 0.6666666665) equal to 1.56347192 and -1.56347192.Subproblem 1
n + 0.6666666665 = 1.56347192 Simplifying n + 0.6666666665 = 1.56347192 Reorder the terms: 0.6666666665 + n = 1.56347192 Solving 0.6666666665 + n = 1.56347192 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 = 1.56347192 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = 1.56347192 + -0.6666666665 n = 1.56347192 + -0.6666666665 Combine like terms: 1.56347192 + -0.6666666665 = 0.8968052535 n = 0.8968052535 Simplifying n = 0.8968052535Subproblem 2
n + 0.6666666665 = -1.56347192 Simplifying n + 0.6666666665 = -1.56347192 Reorder the terms: 0.6666666665 + n = -1.56347192 Solving 0.6666666665 + n = -1.56347192 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 = -1.56347192 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + n = -1.56347192 + -0.6666666665 n = -1.56347192 + -0.6666666665 Combine like terms: -1.56347192 + -0.6666666665 = -2.2301385865 n = -2.2301385865 Simplifying n = -2.2301385865Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.8968052535, -2.2301385865}
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