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Simplifying (2n) * 4n + 40 + -1n + 200 = 400 Remove parenthesis around (2n) 2n * 4n + 40 + -1n + 200 = 400 Reorder the terms for easier multiplication: 2 * 4n * n + 40 + -1n + 200 = 400 Multiply 2 * 4 8n * n + 40 + -1n + 200 = 400 Multiply n * n 8n2 + 40 + -1n + 200 = 400 Reorder the terms: 40 + 200 + -1n + 8n2 = 400 Combine like terms: 40 + 200 = 240 240 + -1n + 8n2 = 400 Solving 240 + -1n + 8n2 = 400 Solving for variable 'n'. Reorder the terms: 240 + -400 + -1n + 8n2 = 400 + -400 Combine like terms: 240 + -400 = -160 -160 + -1n + 8n2 = 400 + -400 Combine like terms: 400 + -400 = 0 -160 + -1n + 8n2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -20 + -0.125n + n2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + -0.125n + 20 + n2 = 0 + 20 Reorder the terms: -20 + 20 + -0.125n + n2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + -0.125n + n2 = 0 + 20 -0.125n + n2 = 0 + 20 Combine like terms: 0 + 20 = 20 -0.125n + n2 = 20 The n term is -0.125n. Take half its coefficient (-0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. -0.125n + 0.00390625 + n2 = 20 + 0.00390625 Reorder the terms: 0.00390625 + -0.125n + n2 = 20 + 0.00390625 Combine like terms: 20 + 0.00390625 = 20.00390625 0.00390625 + -0.125n + n2 = 20.00390625 Factor a perfect square on the left side: (n + -0.0625)(n + -0.0625) = 20.00390625 Calculate the square root of the right side: 4.472572666 Break this problem into two subproblems by setting (n + -0.0625) equal to 4.472572666 and -4.472572666.Subproblem 1
n + -0.0625 = 4.472572666 Simplifying n + -0.0625 = 4.472572666 Reorder the terms: -0.0625 + n = 4.472572666 Solving -0.0625 + n = 4.472572666 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = 4.472572666 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = 4.472572666 + 0.0625 n = 4.472572666 + 0.0625 Combine like terms: 4.472572666 + 0.0625 = 4.535072666 n = 4.535072666 Simplifying n = 4.535072666Subproblem 2
n + -0.0625 = -4.472572666 Simplifying n + -0.0625 = -4.472572666 Reorder the terms: -0.0625 + n = -4.472572666 Solving -0.0625 + n = -4.472572666 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + n = -4.472572666 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + n = -4.472572666 + 0.0625 n = -4.472572666 + 0.0625 Combine like terms: -4.472572666 + 0.0625 = -4.410072666 n = -4.410072666 Simplifying n = -4.410072666Solution
The solution to the problem is based on the solutions from the subproblems. n = {4.535072666, -4.410072666}
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