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Simplifying 4n2 + -11n = 11 Reorder the terms: -11n + 4n2 = 11 Solving -11n + 4n2 = 11 Solving for variable 'n'. Reorder the terms: -11 + -11n + 4n2 = 11 + -11 Combine like terms: 11 + -11 = 0 -11 + -11n + 4n2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.75 + -2.75n + n2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + -2.75n + 2.75 + n2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + -2.75n + n2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + -2.75n + n2 = 0 + 2.75 -2.75n + n2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 -2.75n + n2 = 2.75 The n term is -2.75n. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75n + 1.890625 + n2 = 2.75 + 1.890625 Reorder the terms: 1.890625 + -2.75n + n2 = 2.75 + 1.890625 Combine like terms: 2.75 + 1.890625 = 4.640625 1.890625 + -2.75n + n2 = 4.640625 Factor a perfect square on the left side: (n + -1.375)(n + -1.375) = 4.640625 Calculate the square root of the right side: 2.154210992 Break this problem into two subproblems by setting (n + -1.375) equal to 2.154210992 and -2.154210992.Subproblem 1
n + -1.375 = 2.154210992 Simplifying n + -1.375 = 2.154210992 Reorder the terms: -1.375 + n = 2.154210992 Solving -1.375 + n = 2.154210992 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + n = 2.154210992 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + n = 2.154210992 + 1.375 n = 2.154210992 + 1.375 Combine like terms: 2.154210992 + 1.375 = 3.529210992 n = 3.529210992 Simplifying n = 3.529210992Subproblem 2
n + -1.375 = -2.154210992 Simplifying n + -1.375 = -2.154210992 Reorder the terms: -1.375 + n = -2.154210992 Solving -1.375 + n = -2.154210992 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + n = -2.154210992 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + n = -2.154210992 + 1.375 n = -2.154210992 + 1.375 Combine like terms: -2.154210992 + 1.375 = -0.779210992 n = -0.779210992 Simplifying n = -0.779210992Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.529210992, -0.779210992}
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