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Simplifying 14n2 + -18n + -16 = 5n Reorder the terms: -16 + -18n + 14n2 = 5n Solving -16 + -18n + 14n2 = 5n Solving for variable 'n'. Reorder the terms: -16 + -18n + -5n + 14n2 = 5n + -5n Combine like terms: -18n + -5n = -23n -16 + -23n + 14n2 = 5n + -5n Combine like terms: 5n + -5n = 0 -16 + -23n + 14n2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -1.142857143 + -1.642857143n + n2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + -1.642857143n + 1.142857143 + n2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + -1.642857143n + n2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + -1.642857143n + n2 = 0 + 1.142857143 -1.642857143n + n2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 -1.642857143n + n2 = 1.142857143 The n term is -1.642857143n. Take half its coefficient (-0.8214285715). Square it (0.6747448981) and add it to both sides. Add '0.6747448981' to each side of the equation. -1.642857143n + 0.6747448981 + n2 = 1.142857143 + 0.6747448981 Reorder the terms: 0.6747448981 + -1.642857143n + n2 = 1.142857143 + 0.6747448981 Combine like terms: 1.142857143 + 0.6747448981 = 1.8176020411 0.6747448981 + -1.642857143n + n2 = 1.8176020411 Factor a perfect square on the left side: (n + -0.8214285715)(n + -0.8214285715) = 1.8176020411 Calculate the square root of the right side: 1.348184721 Break this problem into two subproblems by setting (n + -0.8214285715) equal to 1.348184721 and -1.348184721.Subproblem 1
n + -0.8214285715 = 1.348184721 Simplifying n + -0.8214285715 = 1.348184721 Reorder the terms: -0.8214285715 + n = 1.348184721 Solving -0.8214285715 + n = 1.348184721 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8214285715' to each side of the equation. -0.8214285715 + 0.8214285715 + n = 1.348184721 + 0.8214285715 Combine like terms: -0.8214285715 + 0.8214285715 = 0.0000000000 0.0000000000 + n = 1.348184721 + 0.8214285715 n = 1.348184721 + 0.8214285715 Combine like terms: 1.348184721 + 0.8214285715 = 2.1696132925 n = 2.1696132925 Simplifying n = 2.1696132925Subproblem 2
n + -0.8214285715 = -1.348184721 Simplifying n + -0.8214285715 = -1.348184721 Reorder the terms: -0.8214285715 + n = -1.348184721 Solving -0.8214285715 + n = -1.348184721 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8214285715' to each side of the equation. -0.8214285715 + 0.8214285715 + n = -1.348184721 + 0.8214285715 Combine like terms: -0.8214285715 + 0.8214285715 = 0.0000000000 0.0000000000 + n = -1.348184721 + 0.8214285715 n = -1.348184721 + 0.8214285715 Combine like terms: -1.348184721 + 0.8214285715 = -0.5267561495 n = -0.5267561495 Simplifying n = -0.5267561495Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.1696132925, -0.5267561495}
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