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Simplifying 3n + -2n(n + -7) = 3 + 2n Reorder the terms: 3n + -2n(-7 + n) = 3 + 2n 3n + (-7 * -2n + n * -2n) = 3 + 2n 3n + (14n + -2n2) = 3 + 2n Combine like terms: 3n + 14n = 17n 17n + -2n2 = 3 + 2n Solving 17n + -2n2 = 3 + 2n Solving for variable 'n'. Reorder the terms: -3 + 17n + -2n + -2n2 = 3 + 2n + -3 + -2n Combine like terms: 17n + -2n = 15n -3 + 15n + -2n2 = 3 + 2n + -3 + -2n Reorder the terms: -3 + 15n + -2n2 = 3 + -3 + 2n + -2n Combine like terms: 3 + -3 = 0 -3 + 15n + -2n2 = 0 + 2n + -2n -3 + 15n + -2n2 = 2n + -2n Combine like terms: 2n + -2n = 0 -3 + 15n + -2n2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 1.5 + -7.5n + n2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -7.5n + -1.5 + n2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -7.5n + n2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -7.5n + n2 = 0 + -1.5 -7.5n + n2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -7.5n + n2 = -1.5 The n term is -7.5n. Take half its coefficient (-3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. -7.5n + 14.0625 + n2 = -1.5 + 14.0625 Reorder the terms: 14.0625 + -7.5n + n2 = -1.5 + 14.0625 Combine like terms: -1.5 + 14.0625 = 12.5625 14.0625 + -7.5n + n2 = 12.5625 Factor a perfect square on the left side: (n + -3.75)(n + -3.75) = 12.5625 Calculate the square root of the right side: 3.54436172 Break this problem into two subproblems by setting (n + -3.75) equal to 3.54436172 and -3.54436172.Subproblem 1
n + -3.75 = 3.54436172 Simplifying n + -3.75 = 3.54436172 Reorder the terms: -3.75 + n = 3.54436172 Solving -3.75 + n = 3.54436172 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + n = 3.54436172 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + n = 3.54436172 + 3.75 n = 3.54436172 + 3.75 Combine like terms: 3.54436172 + 3.75 = 7.29436172 n = 7.29436172 Simplifying n = 7.29436172Subproblem 2
n + -3.75 = -3.54436172 Simplifying n + -3.75 = -3.54436172 Reorder the terms: -3.75 + n = -3.54436172 Solving -3.75 + n = -3.54436172 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.75' to each side of the equation. -3.75 + 3.75 + n = -3.54436172 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + n = -3.54436172 + 3.75 n = -3.54436172 + 3.75 Combine like terms: -3.54436172 + 3.75 = 0.20563828 n = 0.20563828 Simplifying n = 0.20563828Solution
The solution to the problem is based on the solutions from the subproblems. n = {7.29436172, 0.20563828}
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