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Simplifying (10 + -3.2n) * n = 2 Reorder the terms for easier multiplication: n(10 + -3.2n) = 2 (10 * n + -3.2n * n) = 2 (10n + -3.2n2) = 2 Solving 10n + -3.2n2 = 2 Solving for variable 'n'. Reorder the terms: -2 + 10n + -3.2n2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + 10n + -3.2n2 = 0 Begin completing the square. Divide all terms by -3.2 the coefficient of the squared term: Divide each side by '-3.2'. 0.625 + -3.125n + n2 = 0 Move the constant term to the right: Add '-0.625' to each side of the equation. 0.625 + -3.125n + -0.625 + n2 = 0 + -0.625 Reorder the terms: 0.625 + -0.625 + -3.125n + n2 = 0 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + -3.125n + n2 = 0 + -0.625 -3.125n + n2 = 0 + -0.625 Combine like terms: 0 + -0.625 = -0.625 -3.125n + n2 = -0.625 The n term is -3.125n. Take half its coefficient (-1.5625). Square it (2.44140625) and add it to both sides. Add '2.44140625' to each side of the equation. -3.125n + 2.44140625 + n2 = -0.625 + 2.44140625 Reorder the terms: 2.44140625 + -3.125n + n2 = -0.625 + 2.44140625 Combine like terms: -0.625 + 2.44140625 = 1.81640625 2.44140625 + -3.125n + n2 = 1.81640625 Factor a perfect square on the left side: (n + -1.5625)(n + -1.5625) = 1.81640625 Calculate the square root of the right side: 1.347741166 Break this problem into two subproblems by setting (n + -1.5625) equal to 1.347741166 and -1.347741166.Subproblem 1
n + -1.5625 = 1.347741166 Simplifying n + -1.5625 = 1.347741166 Reorder the terms: -1.5625 + n = 1.347741166 Solving -1.5625 + n = 1.347741166 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + n = 1.347741166 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + n = 1.347741166 + 1.5625 n = 1.347741166 + 1.5625 Combine like terms: 1.347741166 + 1.5625 = 2.910241166 n = 2.910241166 Simplifying n = 2.910241166Subproblem 2
n + -1.5625 = -1.347741166 Simplifying n + -1.5625 = -1.347741166 Reorder the terms: -1.5625 + n = -1.347741166 Solving -1.5625 + n = -1.347741166 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.5625' to each side of the equation. -1.5625 + 1.5625 + n = -1.347741166 + 1.5625 Combine like terms: -1.5625 + 1.5625 = 0.0000 0.0000 + n = -1.347741166 + 1.5625 n = -1.347741166 + 1.5625 Combine like terms: -1.347741166 + 1.5625 = 0.214758834 n = 0.214758834 Simplifying n = 0.214758834Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.910241166, 0.214758834}
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