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Simplifying 10n(n + 5) = 3n + 7 + -2n Reorder the terms: 10n(5 + n) = 3n + 7 + -2n (5 * 10n + n * 10n) = 3n + 7 + -2n (50n + 10n2) = 3n + 7 + -2n Reorder the terms: 50n + 10n2 = 7 + 3n + -2n Combine like terms: 3n + -2n = 1n 50n + 10n2 = 7 + 1n Solving 50n + 10n2 = 7 + 1n Solving for variable 'n'. Reorder the terms: -7 + 50n + -1n + 10n2 = 7 + 1n + -7 + -1n Combine like terms: 50n + -1n = 49n -7 + 49n + 10n2 = 7 + 1n + -7 + -1n Reorder the terms: -7 + 49n + 10n2 = 7 + -7 + 1n + -1n Combine like terms: 7 + -7 = 0 -7 + 49n + 10n2 = 0 + 1n + -1n -7 + 49n + 10n2 = 1n + -1n Combine like terms: 1n + -1n = 0 -7 + 49n + 10n2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.7 + 4.9n + n2 = 0 Move the constant term to the right: Add '0.7' to each side of the equation. -0.7 + 4.9n + 0.7 + n2 = 0 + 0.7 Reorder the terms: -0.7 + 0.7 + 4.9n + n2 = 0 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + 4.9n + n2 = 0 + 0.7 4.9n + n2 = 0 + 0.7 Combine like terms: 0 + 0.7 = 0.7 4.9n + n2 = 0.7 The n term is 4.9n. Take half its coefficient (2.45). Square it (6.0025) and add it to both sides. Add '6.0025' to each side of the equation. 4.9n + 6.0025 + n2 = 0.7 + 6.0025 Reorder the terms: 6.0025 + 4.9n + n2 = 0.7 + 6.0025 Combine like terms: 0.7 + 6.0025 = 6.7025 6.0025 + 4.9n + n2 = 6.7025 Factor a perfect square on the left side: (n + 2.45)(n + 2.45) = 6.7025 Calculate the square root of the right side: 2.588918693 Break this problem into two subproblems by setting (n + 2.45) equal to 2.588918693 and -2.588918693.Subproblem 1
n + 2.45 = 2.588918693 Simplifying n + 2.45 = 2.588918693 Reorder the terms: 2.45 + n = 2.588918693 Solving 2.45 + n = 2.588918693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.45' to each side of the equation. 2.45 + -2.45 + n = 2.588918693 + -2.45 Combine like terms: 2.45 + -2.45 = 0.00 0.00 + n = 2.588918693 + -2.45 n = 2.588918693 + -2.45 Combine like terms: 2.588918693 + -2.45 = 0.138918693 n = 0.138918693 Simplifying n = 0.138918693Subproblem 2
n + 2.45 = -2.588918693 Simplifying n + 2.45 = -2.588918693 Reorder the terms: 2.45 + n = -2.588918693 Solving 2.45 + n = -2.588918693 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.45' to each side of the equation. 2.45 + -2.45 + n = -2.588918693 + -2.45 Combine like terms: 2.45 + -2.45 = 0.00 0.00 + n = -2.588918693 + -2.45 n = -2.588918693 + -2.45 Combine like terms: -2.588918693 + -2.45 = -5.038918693 n = -5.038918693 Simplifying n = -5.038918693Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.138918693, -5.038918693}
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