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Simplifying 12n2 + -12n = 11 Reorder the terms: -12n + 12n2 = 11 Solving -12n + 12n2 = 11 Solving for variable 'n'. Reorder the terms: -11 + -12n + 12n2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + -12n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.9166666667 + -1n + n2 = 0 Move the constant term to the right: Add '0.9166666667' to each side of the equation. -0.9166666667 + -1n + 0.9166666667 + n2 = 0 + 0.9166666667 Reorder the terms: -0.9166666667 + 0.9166666667 + -1n + n2 = 0 + 0.9166666667 Combine like terms: -0.9166666667 + 0.9166666667 = 0.0000000000 0.0000000000 + -1n + n2 = 0 + 0.9166666667 -1n + n2 = 0 + 0.9166666667 Combine like terms: 0 + 0.9166666667 = 0.9166666667 -1n + n2 = 0.9166666667 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 = 0.9166666667 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 0.9166666667 + 0.25 Combine like terms: 0.9166666667 + 0.25 = 1.1666666667 0.25 + -1n + n2 = 1.1666666667 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 1.1666666667 Calculate the square root of the right side: 1.08012345 Break this problem into two subproblems by setting (n + -0.5) equal to 1.08012345 and -1.08012345.Subproblem 1
n + -0.5 = 1.08012345 Simplifying n + -0.5 = 1.08012345 Reorder the terms: -0.5 + n = 1.08012345 Solving -0.5 + n = 1.08012345 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 = 1.08012345 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 1.08012345 + 0.5 n = 1.08012345 + 0.5 Combine like terms: 1.08012345 + 0.5 = 1.58012345 n = 1.58012345 Simplifying n = 1.58012345Subproblem 2
n + -0.5 = -1.08012345 Simplifying n + -0.5 = -1.08012345 Reorder the terms: -0.5 + n = -1.08012345 Solving -0.5 + n = -1.08012345 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 = -1.08012345 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -1.08012345 + 0.5 n = -1.08012345 + 0.5 Combine like terms: -1.08012345 + 0.5 = -0.58012345 n = -0.58012345 Simplifying n = -0.58012345Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.58012345, -0.58012345}
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