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Simplifying 19n2 + 13n + -24 = 6n + 8n2 Reorder the terms: -24 + 13n + 19n2 = 6n + 8n2 Solving -24 + 13n + 19n2 = 6n + 8n2 Solving for variable 'n'. Reorder the terms: -24 + 13n + -6n + 19n2 + -8n2 = 6n + 8n2 + -6n + -8n2 Combine like terms: 13n + -6n = 7n -24 + 7n + 19n2 + -8n2 = 6n + 8n2 + -6n + -8n2 Combine like terms: 19n2 + -8n2 = 11n2 -24 + 7n + 11n2 = 6n + 8n2 + -6n + -8n2 Reorder the terms: -24 + 7n + 11n2 = 6n + -6n + 8n2 + -8n2 Combine like terms: 6n + -6n = 0 -24 + 7n + 11n2 = 0 + 8n2 + -8n2 -24 + 7n + 11n2 = 8n2 + -8n2 Combine like terms: 8n2 + -8n2 = 0 -24 + 7n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -2.181818182 + 0.6363636364n + n2 = 0 Move the constant term to the right: Add '2.181818182' to each side of the equation. -2.181818182 + 0.6363636364n + 2.181818182 + n2 = 0 + 2.181818182 Reorder the terms: -2.181818182 + 2.181818182 + 0.6363636364n + n2 = 0 + 2.181818182 Combine like terms: -2.181818182 + 2.181818182 = 0.000000000 0.000000000 + 0.6363636364n + n2 = 0 + 2.181818182 0.6363636364n + n2 = 0 + 2.181818182 Combine like terms: 0 + 2.181818182 = 2.181818182 0.6363636364n + n2 = 2.181818182 The n term is 0.6363636364n. Take half its coefficient (0.3181818182). Square it (0.1012396694) and add it to both sides. Add '0.1012396694' to each side of the equation. 0.6363636364n + 0.1012396694 + n2 = 2.181818182 + 0.1012396694 Reorder the terms: 0.1012396694 + 0.6363636364n + n2 = 2.181818182 + 0.1012396694 Combine like terms: 2.181818182 + 0.1012396694 = 2.2830578514 0.1012396694 + 0.6363636364n + n2 = 2.2830578514 Factor a perfect square on the left side: (n + 0.3181818182)(n + 0.3181818182) = 2.2830578514 Calculate the square root of the right side: 1.510979104 Break this problem into two subproblems by setting (n + 0.3181818182) equal to 1.510979104 and -1.510979104.Subproblem 1
n + 0.3181818182 = 1.510979104 Simplifying n + 0.3181818182 = 1.510979104 Reorder the terms: 0.3181818182 + n = 1.510979104 Solving 0.3181818182 + n = 1.510979104 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3181818182' to each side of the equation. 0.3181818182 + -0.3181818182 + n = 1.510979104 + -0.3181818182 Combine like terms: 0.3181818182 + -0.3181818182 = 0.0000000000 0.0000000000 + n = 1.510979104 + -0.3181818182 n = 1.510979104 + -0.3181818182 Combine like terms: 1.510979104 + -0.3181818182 = 1.1927972858 n = 1.1927972858 Simplifying n = 1.1927972858Subproblem 2
n + 0.3181818182 = -1.510979104 Simplifying n + 0.3181818182 = -1.510979104 Reorder the terms: 0.3181818182 + n = -1.510979104 Solving 0.3181818182 + n = -1.510979104 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3181818182' to each side of the equation. 0.3181818182 + -0.3181818182 + n = -1.510979104 + -0.3181818182 Combine like terms: 0.3181818182 + -0.3181818182 = 0.0000000000 0.0000000000 + n = -1.510979104 + -0.3181818182 n = -1.510979104 + -0.3181818182 Combine like terms: -1.510979104 + -0.3181818182 = -1.8291609222 n = -1.8291609222 Simplifying n = -1.8291609222Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.1927972858, -1.8291609222}
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