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Simplifying (21 + n) * n = 725 Reorder the terms for easier multiplication: n(21 + n) = 725 (21 * n + n * n) = 725 (21n + n2) = 725 Solving 21n + n2 = 725 Solving for variable 'n'. Reorder the terms: -725 + 21n + n2 = 725 + -725 Combine like terms: 725 + -725 = 0 -725 + 21n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '725' to each side of the equation. -725 + 21n + 725 + n2 = 0 + 725 Reorder the terms: -725 + 725 + 21n + n2 = 0 + 725 Combine like terms: -725 + 725 = 0 0 + 21n + n2 = 0 + 725 21n + n2 = 0 + 725 Combine like terms: 0 + 725 = 725 21n + n2 = 725 The n term is 21n. Take half its coefficient (10.5). Square it (110.25) and add it to both sides. Add '110.25' to each side of the equation. 21n + 110.25 + n2 = 725 + 110.25 Reorder the terms: 110.25 + 21n + n2 = 725 + 110.25 Combine like terms: 725 + 110.25 = 835.25 110.25 + 21n + n2 = 835.25 Factor a perfect square on the left side: (n + 10.5)(n + 10.5) = 835.25 Calculate the square root of the right side: 28.900692033 Break this problem into two subproblems by setting (n + 10.5) equal to 28.900692033 and -28.900692033.Subproblem 1
n + 10.5 = 28.900692033 Simplifying n + 10.5 = 28.900692033 Reorder the terms: 10.5 + n = 28.900692033 Solving 10.5 + n = 28.900692033 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.5' to each side of the equation. 10.5 + -10.5 + n = 28.900692033 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + n = 28.900692033 + -10.5 n = 28.900692033 + -10.5 Combine like terms: 28.900692033 + -10.5 = 18.400692033 n = 18.400692033 Simplifying n = 18.400692033Subproblem 2
n + 10.5 = -28.900692033 Simplifying n + 10.5 = -28.900692033 Reorder the terms: 10.5 + n = -28.900692033 Solving 10.5 + n = -28.900692033 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-10.5' to each side of the equation. 10.5 + -10.5 + n = -28.900692033 + -10.5 Combine like terms: 10.5 + -10.5 = 0.0 0.0 + n = -28.900692033 + -10.5 n = -28.900692033 + -10.5 Combine like terms: -28.900692033 + -10.5 = -39.400692033 n = -39.400692033 Simplifying n = -39.400692033Solution
The solution to the problem is based on the solutions from the subproblems. n = {18.400692033, -39.400692033}
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