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Simplifying 10 + n + 2(n + 10) * n + 10 = 130 Reorder the terms: 10 + n + 2(10 + n) * n + 10 = 130 Reorder the terms for easier multiplication: 10 + n + 2n(10 + n) + 10 = 130 10 + n + (10 * 2n + n * 2n) + 10 = 130 10 + n + (20n + 2n2) + 10 = 130 Reorder the terms: 10 + 10 + n + 20n + 2n2 = 130 Combine like terms: 10 + 10 = 20 20 + n + 20n + 2n2 = 130 Combine like terms: n + 20n = 21n 20 + 21n + 2n2 = 130 Solving 20 + 21n + 2n2 = 130 Solving for variable 'n'. Reorder the terms: 20 + -130 + 21n + 2n2 = 130 + -130 Combine like terms: 20 + -130 = -110 -110 + 21n + 2n2 = 130 + -130 Combine like terms: 130 + -130 = 0 -110 + 21n + 2n2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -55 + 10.5n + n2 = 0 Move the constant term to the right: Add '55' to each side of the equation. -55 + 10.5n + 55 + n2 = 0 + 55 Reorder the terms: -55 + 55 + 10.5n + n2 = 0 + 55 Combine like terms: -55 + 55 = 0 0 + 10.5n + n2 = 0 + 55 10.5n + n2 = 0 + 55 Combine like terms: 0 + 55 = 55 10.5n + n2 = 55 The n term is 10.5n. Take half its coefficient (5.25). Square it (27.5625) and add it to both sides. Add '27.5625' to each side of the equation. 10.5n + 27.5625 + n2 = 55 + 27.5625 Reorder the terms: 27.5625 + 10.5n + n2 = 55 + 27.5625 Combine like terms: 55 + 27.5625 = 82.5625 27.5625 + 10.5n + n2 = 82.5625 Factor a perfect square on the left side: (n + 5.25)(n + 5.25) = 82.5625 Calculate the square root of the right side: 9.086390923 Break this problem into two subproblems by setting (n + 5.25) equal to 9.086390923 and -9.086390923.Subproblem 1
n + 5.25 = 9.086390923 Simplifying n + 5.25 = 9.086390923 Reorder the terms: 5.25 + n = 9.086390923 Solving 5.25 + n = 9.086390923 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + n = 9.086390923 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + n = 9.086390923 + -5.25 n = 9.086390923 + -5.25 Combine like terms: 9.086390923 + -5.25 = 3.836390923 n = 3.836390923 Simplifying n = 3.836390923Subproblem 2
n + 5.25 = -9.086390923 Simplifying n + 5.25 = -9.086390923 Reorder the terms: 5.25 + n = -9.086390923 Solving 5.25 + n = -9.086390923 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-5.25' to each side of the equation. 5.25 + -5.25 + n = -9.086390923 + -5.25 Combine like terms: 5.25 + -5.25 = 0.00 0.00 + n = -9.086390923 + -5.25 n = -9.086390923 + -5.25 Combine like terms: -9.086390923 + -5.25 = -14.336390923 n = -14.336390923 Simplifying n = -14.336390923Solution
The solution to the problem is based on the solutions from the subproblems. n = {3.836390923, -14.336390923}
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