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Simplifying 0.2 = 0.2n(n) + 7.2n + 1.5 Multiply n * n 0.2 = 0.2n2 + 7.2n + 1.5 Reorder the terms: 0.2 = 1.5 + 7.2n + 0.2n2 Solving 0.2 = 1.5 + 7.2n + 0.2n2 Solving for variable 'n'. Combine like terms: 0.2 + -1.5 = -1.3 -1.3 + -7.2n + -0.2n2 = 1.5 + 7.2n + 0.2n2 + -1.5 + -7.2n + -0.2n2 Reorder the terms: -1.3 + -7.2n + -0.2n2 = 1.5 + -1.5 + 7.2n + -7.2n + 0.2n2 + -0.2n2 Combine like terms: 1.5 + -1.5 = 0.0 -1.3 + -7.2n + -0.2n2 = 0.0 + 7.2n + -7.2n + 0.2n2 + -0.2n2 -1.3 + -7.2n + -0.2n2 = 7.2n + -7.2n + 0.2n2 + -0.2n2 Combine like terms: 7.2n + -7.2n = 0.0 -1.3 + -7.2n + -0.2n2 = 0.0 + 0.2n2 + -0.2n2 -1.3 + -7.2n + -0.2n2 = 0.2n2 + -0.2n2 Combine like terms: 0.2n2 + -0.2n2 = 0.0 -1.3 + -7.2n + -0.2n2 = 0.0 Begin completing the square. Divide all terms by -0.2 the coefficient of the squared term: Divide each side by '-0.2'. 6.5 + 36n + n2 = 0 Move the constant term to the right: Add '-6.5' to each side of the equation. 6.5 + 36n + -6.5 + n2 = 0 + -6.5 Reorder the terms: 6.5 + -6.5 + 36n + n2 = 0 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + 36n + n2 = 0 + -6.5 36n + n2 = 0 + -6.5 Combine like terms: 0 + -6.5 = -6.5 36n + n2 = -6.5 The n term is 36n. Take half its coefficient (18). Square it (324) and add it to both sides. Add '324' to each side of the equation. 36n + 324 + n2 = -6.5 + 324 Reorder the terms: 324 + 36n + n2 = -6.5 + 324 Combine like terms: -6.5 + 324 = 317.5 324 + 36n + n2 = 317.5 Factor a perfect square on the left side: (n + 18)(n + 18) = 317.5 Calculate the square root of the right side: 17.818529681 Break this problem into two subproblems by setting (n + 18) equal to 17.818529681 and -17.818529681.Subproblem 1
n + 18 = 17.818529681 Simplifying n + 18 = 17.818529681 Reorder the terms: 18 + n = 17.818529681 Solving 18 + n = 17.818529681 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-18' to each side of the equation. 18 + -18 + n = 17.818529681 + -18 Combine like terms: 18 + -18 = 0 0 + n = 17.818529681 + -18 n = 17.818529681 + -18 Combine like terms: 17.818529681 + -18 = -0.181470319 n = -0.181470319 Simplifying n = -0.181470319Subproblem 2
n + 18 = -17.818529681 Simplifying n + 18 = -17.818529681 Reorder the terms: 18 + n = -17.818529681 Solving 18 + n = -17.818529681 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-18' to each side of the equation. 18 + -18 + n = -17.818529681 + -18 Combine like terms: 18 + -18 = 0 0 + n = -17.818529681 + -18 n = -17.818529681 + -18 Combine like terms: -17.818529681 + -18 = -35.818529681 n = -35.818529681 Simplifying n = -35.818529681Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.181470319, -35.818529681}
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