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Simplifying 2n2 = 12n + -1 Reorder the terms: 2n2 = -1 + 12n Solving 2n2 = -1 + 12n Solving for variable 'n'. Reorder the terms: 1 + -12n + 2n2 = -1 + 12n + 1 + -12n Reorder the terms: 1 + -12n + 2n2 = -1 + 1 + 12n + -12n Combine like terms: -1 + 1 = 0 1 + -12n + 2n2 = 0 + 12n + -12n 1 + -12n + 2n2 = 12n + -12n Combine like terms: 12n + -12n = 0 1 + -12n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -6n + n2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -6n + -0.5 + n2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -6n + n2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -6n + n2 = 0 + -0.5 -6n + n2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -6n + n2 = -0.5 The n term is -6n. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6n + 9 + n2 = -0.5 + 9 Reorder the terms: 9 + -6n + n2 = -0.5 + 9 Combine like terms: -0.5 + 9 = 8.5 9 + -6n + n2 = 8.5 Factor a perfect square on the left side: (n + -3)(n + -3) = 8.5 Calculate the square root of the right side: 2.915475947 Break this problem into two subproblems by setting (n + -3) equal to 2.915475947 and -2.915475947.Subproblem 1
n + -3 = 2.915475947 Simplifying n + -3 = 2.915475947 Reorder the terms: -3 + n = 2.915475947 Solving -3 + n = 2.915475947 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.915475947 + 3 n = 2.915475947 + 3 Combine like terms: 2.915475947 + 3 = 5.915475947 n = 5.915475947 Simplifying n = 5.915475947Subproblem 2
n + -3 = -2.915475947 Simplifying n + -3 = -2.915475947 Reorder the terms: -3 + n = -2.915475947 Solving -3 + n = -2.915475947 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -2.915475947 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.915475947 + 3 n = -2.915475947 + 3 Combine like terms: -2.915475947 + 3 = 0.084524053 n = 0.084524053 Simplifying n = 0.084524053Solution
The solution to the problem is based on the solutions from the subproblems. n = {5.915475947, 0.084524053}
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