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Simplifying (2n2 + -5n + -6) + (-1n + -3n + 11) = 0 Reorder the terms: (-6 + -5n + 2n2) + (-1n + -3n + 11) = 0 Remove parenthesis around (-6 + -5n + 2n2) -6 + -5n + 2n2 + (-1n + -3n + 11) = 0 Reorder the terms: -6 + -5n + 2n2 + (11 + -1n + -3n) = 0 Combine like terms: -1n + -3n = -4n -6 + -5n + 2n2 + (11 + -4n) = 0 Remove parenthesis around (11 + -4n) -6 + -5n + 2n2 + 11 + -4n = 0 Reorder the terms: -6 + 11 + -5n + -4n + 2n2 = 0 Combine like terms: -6 + 11 = 5 5 + -5n + -4n + 2n2 = 0 Combine like terms: -5n + -4n = -9n 5 + -9n + 2n2 = 0 Solving 5 + -9n + 2n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + -4.5n + n2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -4.5n + -2.5 + n2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -4.5n + n2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -4.5n + n2 = 0 + -2.5 -4.5n + n2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -4.5n + n2 = -2.5 The n term is -4.5n. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5n + 5.0625 + n2 = -2.5 + 5.0625 Reorder the terms: 5.0625 + -4.5n + n2 = -2.5 + 5.0625 Combine like terms: -2.5 + 5.0625 = 2.5625 5.0625 + -4.5n + n2 = 2.5625 Factor a perfect square on the left side: (n + -2.25)(n + -2.25) = 2.5625 Calculate the square root of the right side: 1.600781059 Break this problem into two subproblems by setting (n + -2.25) equal to 1.600781059 and -1.600781059.Subproblem 1
n + -2.25 = 1.600781059 Simplifying n + -2.25 = 1.600781059 Reorder the terms: -2.25 + n = 1.600781059 Solving -2.25 + n = 1.600781059 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + n = 1.600781059 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = 1.600781059 + 2.25 n = 1.600781059 + 2.25 Combine like terms: 1.600781059 + 2.25 = 3.850781059 n = 3.850781059 Simplifying n = 3.850781059Subproblem 2
n + -2.25 = -1.600781059 Simplifying n + -2.25 = -1.600781059 Reorder the terms: -2.25 + n = -1.600781059 Solving -2.25 + n = -1.600781059 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + n = -1.600781059 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + n = -1.600781059 + 2.25 n = -1.600781059 + 2.25 Combine like terms: -1.600781059 + 2.25 = 0.649218941 n = 0.649218941 Simplifying n = 0.649218941Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.850781059, 0.649218941}
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