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Simplifying 2n2 + -11n + 3 = 19 Reorder the terms: 3 + -11n + 2n2 = 19 Solving 3 + -11n + 2n2 = 19 Solving for variable 'n'. Reorder the terms: 3 + -19 + -11n + 2n2 = 19 + -19 Combine like terms: 3 + -19 = -16 -16 + -11n + 2n2 = 19 + -19 Combine like terms: 19 + -19 = 0 -16 + -11n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8 + -5.5n + n2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -5.5n + 8 + n2 = 0 + 8 Reorder the terms: -8 + 8 + -5.5n + n2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -5.5n + n2 = 0 + 8 -5.5n + n2 = 0 + 8 Combine like terms: 0 + 8 = 8 -5.5n + n2 = 8 The n term is -5.5n. Take half its coefficient (-2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. -5.5n + 7.5625 + n2 = 8 + 7.5625 Reorder the terms: 7.5625 + -5.5n + n2 = 8 + 7.5625 Combine like terms: 8 + 7.5625 = 15.5625 7.5625 + -5.5n + n2 = 15.5625 Factor a perfect square on the left side: (n + -2.75)(n + -2.75) = 15.5625 Calculate the square root of the right side: 3.94493346 Break this problem into two subproblems by setting (n + -2.75) equal to 3.94493346 and -3.94493346.Subproblem 1
n + -2.75 = 3.94493346 Simplifying n + -2.75 = 3.94493346 Reorder the terms: -2.75 + n = 3.94493346 Solving -2.75 + n = 3.94493346 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + n = 3.94493346 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + n = 3.94493346 + 2.75 n = 3.94493346 + 2.75 Combine like terms: 3.94493346 + 2.75 = 6.69493346 n = 6.69493346 Simplifying n = 6.69493346Subproblem 2
n + -2.75 = -3.94493346 Simplifying n + -2.75 = -3.94493346 Reorder the terms: -2.75 + n = -3.94493346 Solving -2.75 + n = -3.94493346 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.75' to each side of the equation. -2.75 + 2.75 + n = -3.94493346 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + n = -3.94493346 + 2.75 n = -3.94493346 + 2.75 Combine like terms: -3.94493346 + 2.75 = -1.19493346 n = -1.19493346 Simplifying n = -1.19493346Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.69493346, -1.19493346}
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