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Simplifying 3n2 + -12 = 7n Reorder the terms: -12 + 3n2 = 7n Solving -12 + 3n2 = 7n Solving for variable 'n'. Reorder the terms: -12 + -7n + 3n2 = 7n + -7n Combine like terms: 7n + -7n = 0 -12 + -7n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -4 + -2.333333333n + n2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + -2.333333333n + 4 + n2 = 0 + 4 Reorder the terms: -4 + 4 + -2.333333333n + n2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + -2.333333333n + n2 = 0 + 4 -2.333333333n + n2 = 0 + 4 Combine like terms: 0 + 4 = 4 -2.333333333n + n2 = 4 The n term is -2.333333333n. Take half its coefficient (-1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. -2.333333333n + 1.361111112 + n2 = 4 + 1.361111112 Reorder the terms: 1.361111112 + -2.333333333n + n2 = 4 + 1.361111112 Combine like terms: 4 + 1.361111112 = 5.361111112 1.361111112 + -2.333333333n + n2 = 5.361111112 Factor a perfect square on the left side: (n + -1.166666667)(n + -1.166666667) = 5.361111112 Calculate the square root of the right side: 2.315407332 Break this problem into two subproblems by setting (n + -1.166666667) equal to 2.315407332 and -2.315407332.Subproblem 1
n + -1.166666667 = 2.315407332 Simplifying n + -1.166666667 = 2.315407332 Reorder the terms: -1.166666667 + n = 2.315407332 Solving -1.166666667 + n = 2.315407332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + n = 2.315407332 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + n = 2.315407332 + 1.166666667 n = 2.315407332 + 1.166666667 Combine like terms: 2.315407332 + 1.166666667 = 3.482073999 n = 3.482073999 Simplifying n = 3.482073999Subproblem 2
n + -1.166666667 = -2.315407332 Simplifying n + -1.166666667 = -2.315407332 Reorder the terms: -1.166666667 + n = -2.315407332 Solving -1.166666667 + n = -2.315407332 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + n = -2.315407332 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + n = -2.315407332 + 1.166666667 n = -2.315407332 + 1.166666667 Combine like terms: -2.315407332 + 1.166666667 = -1.148740665 n = -1.148740665 Simplifying n = -1.148740665Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.482073999, -1.148740665}
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