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Simplifying 5n2 + 19n + -69 = -2 Reorder the terms: -69 + 19n + 5n2 = -2 Solving -69 + 19n + 5n2 = -2 Solving for variable 'n'. Reorder the terms: -69 + 2 + 19n + 5n2 = -2 + 2 Combine like terms: -69 + 2 = -67 -67 + 19n + 5n2 = -2 + 2 Combine like terms: -2 + 2 = 0 -67 + 19n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -13.4 + 3.8n + n2 = 0 Move the constant term to the right: Add '13.4' to each side of the equation. -13.4 + 3.8n + 13.4 + n2 = 0 + 13.4 Reorder the terms: -13.4 + 13.4 + 3.8n + n2 = 0 + 13.4 Combine like terms: -13.4 + 13.4 = 0.0 0.0 + 3.8n + n2 = 0 + 13.4 3.8n + n2 = 0 + 13.4 Combine like terms: 0 + 13.4 = 13.4 3.8n + n2 = 13.4 The n term is 3.8n. Take half its coefficient (1.9). Square it (3.61) and add it to both sides. Add '3.61' to each side of the equation. 3.8n + 3.61 + n2 = 13.4 + 3.61 Reorder the terms: 3.61 + 3.8n + n2 = 13.4 + 3.61 Combine like terms: 13.4 + 3.61 = 17.01 3.61 + 3.8n + n2 = 17.01 Factor a perfect square on the left side: (n + 1.9)(n + 1.9) = 17.01 Calculate the square root of the right side: 4.124318125 Break this problem into two subproblems by setting (n + 1.9) equal to 4.124318125 and -4.124318125.Subproblem 1
n + 1.9 = 4.124318125 Simplifying n + 1.9 = 4.124318125 Reorder the terms: 1.9 + n = 4.124318125 Solving 1.9 + n = 4.124318125 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + n = 4.124318125 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 4.124318125 + -1.9 n = 4.124318125 + -1.9 Combine like terms: 4.124318125 + -1.9 = 2.224318125 n = 2.224318125 Simplifying n = 2.224318125Subproblem 2
n + 1.9 = -4.124318125 Simplifying n + 1.9 = -4.124318125 Reorder the terms: 1.9 + n = -4.124318125 Solving 1.9 + n = -4.124318125 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + n = -4.124318125 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -4.124318125 + -1.9 n = -4.124318125 + -1.9 Combine like terms: -4.124318125 + -1.9 = -6.024318125 n = -6.024318125 Simplifying n = -6.024318125Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.224318125, -6.024318125}
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