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Simplifying 2n2 + -12n = -17 Reorder the terms: -12n + 2n2 = -17 Solving -12n + 2n2 = -17 Solving for variable 'n'. Reorder the terms: 17 + -12n + 2n2 = -17 + 17 Combine like terms: -17 + 17 = 0 17 + -12n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 8.5 + -6n + n2 = 0 Move the constant term to the right: Add '-8.5' to each side of the equation. 8.5 + -6n + -8.5 + n2 = 0 + -8.5 Reorder the terms: 8.5 + -8.5 + -6n + n2 = 0 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + -6n + n2 = 0 + -8.5 -6n + n2 = 0 + -8.5 Combine like terms: 0 + -8.5 = -8.5 -6n + n2 = -8.5 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 = -8.5 + 9 Reorder the terms: 9 + -6n + n2 = -8.5 + 9 Combine like terms: -8.5 + 9 = 0.5 9 + -6n + n2 = 0.5 Factor a perfect square on the left side: (n + -3)(n + -3) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (n + -3) equal to 0.707106781 and -0.707106781.Subproblem 1
n + -3 = 0.707106781 Simplifying n + -3 = 0.707106781 Reorder the terms: -3 + n = 0.707106781 Solving -3 + n = 0.707106781 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 = 0.707106781 + 3 Combine like terms: -3 + 3 = 0 0 + n = 0.707106781 + 3 n = 0.707106781 + 3 Combine like terms: 0.707106781 + 3 = 3.707106781 n = 3.707106781 Simplifying n = 3.707106781Subproblem 2
n + -3 = -0.707106781 Simplifying n + -3 = -0.707106781 Reorder the terms: -3 + n = -0.707106781 Solving -3 + n = -0.707106781 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 = -0.707106781 + 3 Combine like terms: -3 + 3 = 0 0 + n = -0.707106781 + 3 n = -0.707106781 + 3 Combine like terms: -0.707106781 + 3 = 2.292893219 n = 2.292893219 Simplifying n = 2.292893219Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.707106781, 2.292893219}
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