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Simplifying 5n2 + 19n = -2 Reorder the terms: 19n + 5n2 = -2 Solving 19n + 5n2 = -2 Solving for variable 'n'. Reorder the terms: 2 + 19n + 5n2 = -2 + 2 Combine like terms: -2 + 2 = 0 2 + 19n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.4 + 3.8n + n2 = 0 Move the constant term to the right: Add '-0.4' to each side of the equation. 0.4 + 3.8n + -0.4 + n2 = 0 + -0.4 Reorder the terms: 0.4 + -0.4 + 3.8n + n2 = 0 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + 3.8n + n2 = 0 + -0.4 3.8n + n2 = 0 + -0.4 Combine like terms: 0 + -0.4 = -0.4 3.8n + n2 = -0.4 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 = -0.4 + 3.61 Reorder the terms: 3.61 + 3.8n + n2 = -0.4 + 3.61 Combine like terms: -0.4 + 3.61 = 3.21 3.61 + 3.8n + n2 = 3.21 Factor a perfect square on the left side: (n + 1.9)(n + 1.9) = 3.21 Calculate the square root of the right side: 1.791647287 Break this problem into two subproblems by setting (n + 1.9) equal to 1.791647287 and -1.791647287.Subproblem 1
n + 1.9 = 1.791647287 Simplifying n + 1.9 = 1.791647287 Reorder the terms: 1.9 + n = 1.791647287 Solving 1.9 + n = 1.791647287 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 = 1.791647287 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 1.791647287 + -1.9 n = 1.791647287 + -1.9 Combine like terms: 1.791647287 + -1.9 = -0.108352713 n = -0.108352713 Simplifying n = -0.108352713Subproblem 2
n + 1.9 = -1.791647287 Simplifying n + 1.9 = -1.791647287 Reorder the terms: 1.9 + n = -1.791647287 Solving 1.9 + n = -1.791647287 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 = -1.791647287 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -1.791647287 + -1.9 n = -1.791647287 + -1.9 Combine like terms: -1.791647287 + -1.9 = -3.691647287 n = -3.691647287 Simplifying n = -3.691647287Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.108352713, -3.691647287}
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