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Simplifying 3n2 + -3n + -8 = 12 Reorder the terms: -8 + -3n + 3n2 = 12 Solving -8 + -3n + 3n2 = 12 Solving for variable 'n'. Reorder the terms: -8 + -12 + -3n + 3n2 = 12 + -12 Combine like terms: -8 + -12 = -20 -20 + -3n + 3n2 = 12 + -12 Combine like terms: 12 + -12 = 0 -20 + -3n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6.666666667 + -1n + n2 = 0 Move the constant term to the right: Add '6.666666667' to each side of the equation. -6.666666667 + -1n + 6.666666667 + n2 = 0 + 6.666666667 Reorder the terms: -6.666666667 + 6.666666667 + -1n + n2 = 0 + 6.666666667 Combine like terms: -6.666666667 + 6.666666667 = 0.000000000 0.000000000 + -1n + n2 = 0 + 6.666666667 -1n + n2 = 0 + 6.666666667 Combine like terms: 0 + 6.666666667 = 6.666666667 -1n + n2 = 6.666666667 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 6.666666667 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 6.666666667 + 0.25 Combine like terms: 6.666666667 + 0.25 = 6.916666667 0.25 + -1n + n2 = 6.916666667 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 6.916666667 Calculate the square root of the right side: 2.62995564 Break this problem into two subproblems by setting (n + -0.5) equal to 2.62995564 and -2.62995564.Subproblem 1
n + -0.5 = 2.62995564 Simplifying n + -0.5 = 2.62995564 Reorder the terms: -0.5 + n = 2.62995564 Solving -0.5 + n = 2.62995564 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 2.62995564 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 2.62995564 + 0.5 n = 2.62995564 + 0.5 Combine like terms: 2.62995564 + 0.5 = 3.12995564 n = 3.12995564 Simplifying n = 3.12995564Subproblem 2
n + -0.5 = -2.62995564 Simplifying n + -0.5 = -2.62995564 Reorder the terms: -0.5 + n = -2.62995564 Solving -0.5 + n = -2.62995564 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -2.62995564 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -2.62995564 + 0.5 n = -2.62995564 + 0.5 Combine like terms: -2.62995564 + 0.5 = -2.12995564 n = -2.12995564 Simplifying n = -2.12995564Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.12995564, -2.12995564}
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