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Simplifying 5n2 + -12 + 19n = 0 Reorder the terms: -12 + 19n + 5n2 = 0 Solving -12 + 19n + 5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.4 + 3.8n + n2 = 0 Move the constant term to the right: Add '2.4' to each side of the equation. -2.4 + 3.8n + 2.4 + n2 = 0 + 2.4 Reorder the terms: -2.4 + 2.4 + 3.8n + n2 = 0 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + 3.8n + n2 = 0 + 2.4 3.8n + n2 = 0 + 2.4 Combine like terms: 0 + 2.4 = 2.4 3.8n + n2 = 2.4 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 = 2.4 + 3.61 Reorder the terms: 3.61 + 3.8n + n2 = 2.4 + 3.61 Combine like terms: 2.4 + 3.61 = 6.01 3.61 + 3.8n + n2 = 6.01 Factor a perfect square on the left side: (n + 1.9)(n + 1.9) = 6.01 Calculate the square root of the right side: 2.451530134 Break this problem into two subproblems by setting (n + 1.9) equal to 2.451530134 and -2.451530134.Subproblem 1
n + 1.9 = 2.451530134 Simplifying n + 1.9 = 2.451530134 Reorder the terms: 1.9 + n = 2.451530134 Solving 1.9 + n = 2.451530134 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 = 2.451530134 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = 2.451530134 + -1.9 n = 2.451530134 + -1.9 Combine like terms: 2.451530134 + -1.9 = 0.551530134 n = 0.551530134 Simplifying n = 0.551530134Subproblem 2
n + 1.9 = -2.451530134 Simplifying n + 1.9 = -2.451530134 Reorder the terms: 1.9 + n = -2.451530134 Solving 1.9 + n = -2.451530134 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 = -2.451530134 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + n = -2.451530134 + -1.9 n = -2.451530134 + -1.9 Combine like terms: -2.451530134 + -1.9 = -4.351530134 n = -4.351530134 Simplifying n = -4.351530134Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.551530134, -4.351530134}
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