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Simplifying (6 * 70) + -1(6 * 4) = n(70 + -1n) Multiply 6 * 70 (420) + -1(6 * 4) = n(70 + -1n) 420 + -1(6 * 4) = n(70 + -1n) Multiply 6 * 4 420 + -1(24) = n(70 + -1n) Multiply -1 * 24 420 + -24 = n(70 + -1n) Combine like terms: 420 + -24 = 396 396 = n(70 + -1n) 396 = (70 * n + -1n * n) 396 = (70n + -1n2) Solving 396 = 70n + -1n2 Solving for variable 'n'. Reorder the terms: 396 + -70n + n2 = 70n + -70n + -1n2 + n2 Combine like terms: 70n + -70n = 0 396 + -70n + n2 = 0 + -1n2 + n2 396 + -70n + n2 = -1n2 + n2 Combine like terms: -1n2 + n2 = 0 396 + -70n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-396' to each side of the equation. 396 + -70n + -396 + n2 = 0 + -396 Reorder the terms: 396 + -396 + -70n + n2 = 0 + -396 Combine like terms: 396 + -396 = 0 0 + -70n + n2 = 0 + -396 -70n + n2 = 0 + -396 Combine like terms: 0 + -396 = -396 -70n + n2 = -396 The n term is -70n. Take half its coefficient (-35). Square it (1225) and add it to both sides. Add '1225' to each side of the equation. -70n + 1225 + n2 = -396 + 1225 Reorder the terms: 1225 + -70n + n2 = -396 + 1225 Combine like terms: -396 + 1225 = 829 1225 + -70n + n2 = 829 Factor a perfect square on the left side: (n + -35)(n + -35) = 829 Calculate the square root of the right side: 28.792360098 Break this problem into two subproblems by setting (n + -35) equal to 28.792360098 and -28.792360098.Subproblem 1
n + -35 = 28.792360098 Simplifying n + -35 = 28.792360098 Reorder the terms: -35 + n = 28.792360098 Solving -35 + n = 28.792360098 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '35' to each side of the equation. -35 + 35 + n = 28.792360098 + 35 Combine like terms: -35 + 35 = 0 0 + n = 28.792360098 + 35 n = 28.792360098 + 35 Combine like terms: 28.792360098 + 35 = 63.792360098 n = 63.792360098 Simplifying n = 63.792360098Subproblem 2
n + -35 = -28.792360098 Simplifying n + -35 = -28.792360098 Reorder the terms: -35 + n = -28.792360098 Solving -35 + n = -28.792360098 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '35' to each side of the equation. -35 + 35 + n = -28.792360098 + 35 Combine like terms: -35 + 35 = 0 0 + n = -28.792360098 + 35 n = -28.792360098 + 35 Combine like terms: -28.792360098 + 35 = 6.207639902 n = 6.207639902 Simplifying n = 6.207639902Solution
The solution to the problem is based on the solutions from the subproblems. n = {63.792360098, 6.207639902}
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