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Simplifying n2 + -5n + 3 = 8n2 + 8n + -4 Reorder the terms: 3 + -5n + n2 = 8n2 + 8n + -4 Reorder the terms: 3 + -5n + n2 = -4 + 8n + 8n2 Solving 3 + -5n + n2 = -4 + 8n + 8n2 Solving for variable 'n'. Reorder the terms: 3 + 4 + -5n + -8n + n2 + -8n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2 Combine like terms: 3 + 4 = 7 7 + -5n + -8n + n2 + -8n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2 Combine like terms: -5n + -8n = -13n 7 + -13n + n2 + -8n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2 Combine like terms: n2 + -8n2 = -7n2 7 + -13n + -7n2 = -4 + 8n + 8n2 + 4 + -8n + -8n2 Reorder the terms: 7 + -13n + -7n2 = -4 + 4 + 8n + -8n + 8n2 + -8n2 Combine like terms: -4 + 4 = 0 7 + -13n + -7n2 = 0 + 8n + -8n + 8n2 + -8n2 7 + -13n + -7n2 = 8n + -8n + 8n2 + -8n2 Combine like terms: 8n + -8n = 0 7 + -13n + -7n2 = 0 + 8n2 + -8n2 7 + -13n + -7n2 = 8n2 + -8n2 Combine like terms: 8n2 + -8n2 = 0 7 + -13n + -7n2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -1 + 1.857142857n + n2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.857142857n + 1 + n2 = 0 + 1 Reorder the terms: -1 + 1 + 1.857142857n + n2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.857142857n + n2 = 0 + 1 1.857142857n + n2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.857142857n + n2 = 1 The n term is 1.857142857n. Take half its coefficient (0.9285714285). Square it (0.8622448978) and add it to both sides. Add '0.8622448978' to each side of the equation. 1.857142857n + 0.8622448978 + n2 = 1 + 0.8622448978 Reorder the terms: 0.8622448978 + 1.857142857n + n2 = 1 + 0.8622448978 Combine like terms: 1 + 0.8622448978 = 1.8622448978 0.8622448978 + 1.857142857n + n2 = 1.8622448978 Factor a perfect square on the left side: (n + 0.9285714285)(n + 0.9285714285) = 1.8622448978 Calculate the square root of the right side: 1.364640941 Break this problem into two subproblems by setting (n + 0.9285714285) equal to 1.364640941 and -1.364640941.Subproblem 1
n + 0.9285714285 = 1.364640941 Simplifying n + 0.9285714285 = 1.364640941 Reorder the terms: 0.9285714285 + n = 1.364640941 Solving 0.9285714285 + n = 1.364640941 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + n = 1.364640941 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + n = 1.364640941 + -0.9285714285 n = 1.364640941 + -0.9285714285 Combine like terms: 1.364640941 + -0.9285714285 = 0.4360695125 n = 0.4360695125 Simplifying n = 0.4360695125Subproblem 2
n + 0.9285714285 = -1.364640941 Simplifying n + 0.9285714285 = -1.364640941 Reorder the terms: 0.9285714285 + n = -1.364640941 Solving 0.9285714285 + n = -1.364640941 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + n = -1.364640941 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + n = -1.364640941 + -0.9285714285 n = -1.364640941 + -0.9285714285 Combine like terms: -1.364640941 + -0.9285714285 = -2.2932123695 n = -2.2932123695 Simplifying n = -2.2932123695Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.4360695125, -2.2932123695}
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