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Simplifying 12n2 + -7n = 108 Reorder the terms: -7n + 12n2 = 108 Solving -7n + 12n2 = 108 Solving for variable 'n'. Reorder the terms: -108 + -7n + 12n2 = 108 + -108 Combine like terms: 108 + -108 = 0 -108 + -7n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -9 + -0.5833333333n + n2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -0.5833333333n + 9 + n2 = 0 + 9 Reorder the terms: -9 + 9 + -0.5833333333n + n2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -0.5833333333n + n2 = 0 + 9 -0.5833333333n + n2 = 0 + 9 Combine like terms: 0 + 9 = 9 -0.5833333333n + n2 = 9 The n term is -0.5833333333n. Take half its coefficient (-0.2916666667). Square it (0.08506944446) and add it to both sides. Add '0.08506944446' to each side of the equation. -0.5833333333n + 0.08506944446 + n2 = 9 + 0.08506944446 Reorder the terms: 0.08506944446 + -0.5833333333n + n2 = 9 + 0.08506944446 Combine like terms: 9 + 0.08506944446 = 9.08506944446 0.08506944446 + -0.5833333333n + n2 = 9.08506944446 Factor a perfect square on the left side: (n + -0.2916666667)(n + -0.2916666667) = 9.08506944446 Calculate the square root of the right side: 3.014144894 Break this problem into two subproblems by setting (n + -0.2916666667) equal to 3.014144894 and -3.014144894.Subproblem 1
n + -0.2916666667 = 3.014144894 Simplifying n + -0.2916666667 = 3.014144894 Reorder the terms: -0.2916666667 + n = 3.014144894 Solving -0.2916666667 + n = 3.014144894 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + n = 3.014144894 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + n = 3.014144894 + 0.2916666667 n = 3.014144894 + 0.2916666667 Combine like terms: 3.014144894 + 0.2916666667 = 3.3058115607 n = 3.3058115607 Simplifying n = 3.3058115607Subproblem 2
n + -0.2916666667 = -3.014144894 Simplifying n + -0.2916666667 = -3.014144894 Reorder the terms: -0.2916666667 + n = -3.014144894 Solving -0.2916666667 + n = -3.014144894 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2916666667' to each side of the equation. -0.2916666667 + 0.2916666667 + n = -3.014144894 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + n = -3.014144894 + 0.2916666667 n = -3.014144894 + 0.2916666667 Combine like terms: -3.014144894 + 0.2916666667 = -2.7224782273 n = -2.7224782273 Simplifying n = -2.7224782273Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.3058115607, -2.7224782273}
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