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Simplifying (3n2 + 7n) + -1(-5n2 + -4n + 3) = 0 Reorder the terms: (7n + 3n2) + -1(-5n2 + -4n + 3) = 0 Remove parenthesis around (7n + 3n2) 7n + 3n2 + -1(-5n2 + -4n + 3) = 0 Reorder the terms: 7n + 3n2 + -1(3 + -4n + -5n2) = 0 7n + 3n2 + (3 * -1 + -4n * -1 + -5n2 * -1) = 0 7n + 3n2 + (-3 + 4n + 5n2) = 0 Reorder the terms: -3 + 7n + 4n + 3n2 + 5n2 = 0 Combine like terms: 7n + 4n = 11n -3 + 11n + 3n2 + 5n2 = 0 Combine like terms: 3n2 + 5n2 = 8n2 -3 + 11n + 8n2 = 0 Solving -3 + 11n + 8n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + 1.375n + n2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + 1.375n + 0.375 + n2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + 1.375n + n2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + 1.375n + n2 = 0 + 0.375 1.375n + n2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 1.375n + n2 = 0.375 The n term is 1.375n. Take half its coefficient (0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. 1.375n + 0.47265625 + n2 = 0.375 + 0.47265625 Reorder the terms: 0.47265625 + 1.375n + n2 = 0.375 + 0.47265625 Combine like terms: 0.375 + 0.47265625 = 0.84765625 0.47265625 + 1.375n + n2 = 0.84765625 Factor a perfect square on the left side: (n + 0.6875)(n + 0.6875) = 0.84765625 Calculate the square root of the right side: 0.920682491 Break this problem into two subproblems by setting (n + 0.6875) equal to 0.920682491 and -0.920682491.Subproblem 1
n + 0.6875 = 0.920682491 Simplifying n + 0.6875 = 0.920682491 Reorder the terms: 0.6875 + n = 0.920682491 Solving 0.6875 + n = 0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + n = 0.920682491 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + n = 0.920682491 + -0.6875 n = 0.920682491 + -0.6875 Combine like terms: 0.920682491 + -0.6875 = 0.233182491 n = 0.233182491 Simplifying n = 0.233182491Subproblem 2
n + 0.6875 = -0.920682491 Simplifying n + 0.6875 = -0.920682491 Reorder the terms: 0.6875 + n = -0.920682491 Solving 0.6875 + n = -0.920682491 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.6875' to each side of the equation. 0.6875 + -0.6875 + n = -0.920682491 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + n = -0.920682491 + -0.6875 n = -0.920682491 + -0.6875 Combine like terms: -0.920682491 + -0.6875 = -1.608182491 n = -1.608182491 Simplifying n = -1.608182491Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.233182491, -1.608182491}
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