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Simplifying 6n2 + -2n + -12 = 0 Reorder the terms: -12 + -2n + 6n2 = 0 Solving -12 + -2n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '2'. 2(-6 + -1n + 3n2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-6 + -1n + 3n2)' equal to zero and attempt to solve: Simplifying -6 + -1n + 3n2 = 0 Solving -6 + -1n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + -0.3333333333n + n2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -0.3333333333n + 2 + n2 = 0 + 2 Reorder the terms: -2 + 2 + -0.3333333333n + n2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -0.3333333333n + n2 = 0 + 2 -0.3333333333n + n2 = 0 + 2 Combine like terms: 0 + 2 = 2 -0.3333333333n + n2 = 2 The n term is -0.3333333333n. Take half its coefficient (-0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. -0.3333333333n + 0.02777777779 + n2 = 2 + 0.02777777779 Reorder the terms: 0.02777777779 + -0.3333333333n + n2 = 2 + 0.02777777779 Combine like terms: 2 + 0.02777777779 = 2.02777777779 0.02777777779 + -0.3333333333n + n2 = 2.02777777779 Factor a perfect square on the left side: (n + -0.1666666667)(n + -0.1666666667) = 2.02777777779 Calculate the square root of the right side: 1.424000624 Break this problem into two subproblems by setting (n + -0.1666666667) equal to 1.424000624 and -1.424000624.Subproblem 1
n + -0.1666666667 = 1.424000624 Simplifying n + -0.1666666667 = 1.424000624 Reorder the terms: -0.1666666667 + n = 1.424000624 Solving -0.1666666667 + n = 1.424000624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + n = 1.424000624 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + n = 1.424000624 + 0.1666666667 n = 1.424000624 + 0.1666666667 Combine like terms: 1.424000624 + 0.1666666667 = 1.5906672907 n = 1.5906672907 Simplifying n = 1.5906672907Subproblem 2
n + -0.1666666667 = -1.424000624 Simplifying n + -0.1666666667 = -1.424000624 Reorder the terms: -0.1666666667 + n = -1.424000624 Solving -0.1666666667 + n = -1.424000624 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + n = -1.424000624 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + n = -1.424000624 + 0.1666666667 n = -1.424000624 + 0.1666666667 Combine like terms: -1.424000624 + 0.1666666667 = -1.2573339573 n = -1.2573339573 Simplifying n = -1.2573339573Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.5906672907, -1.2573339573}Solution
n = {1.5906672907, -1.2573339573}
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