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Simplifying 9r(r + -2) + -1(r + -8) = 0 Reorder the terms: 9r(-2 + r) + -1(r + -8) = 0 (-2 * 9r + r * 9r) + -1(r + -8) = 0 (-18r + 9r2) + -1(r + -8) = 0 Reorder the terms: -18r + 9r2 + -1(-8 + r) = 0 -18r + 9r2 + (-8 * -1 + r * -1) = 0 -18r + 9r2 + (8 + -1r) = 0 Reorder the terms: 8 + -18r + -1r + 9r2 = 0 Combine like terms: -18r + -1r = -19r 8 + -19r + 9r2 = 0 Solving 8 + -19r + 9r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.8888888889 + -2.111111111r + r2 = 0 Move the constant term to the right: Add '-0.8888888889' to each side of the equation. 0.8888888889 + -2.111111111r + -0.8888888889 + r2 = 0 + -0.8888888889 Reorder the terms: 0.8888888889 + -0.8888888889 + -2.111111111r + r2 = 0 + -0.8888888889 Combine like terms: 0.8888888889 + -0.8888888889 = 0.0000000000 0.0000000000 + -2.111111111r + r2 = 0 + -0.8888888889 -2.111111111r + r2 = 0 + -0.8888888889 Combine like terms: 0 + -0.8888888889 = -0.8888888889 -2.111111111r + r2 = -0.8888888889 The r term is -2.111111111r. Take half its coefficient (-1.055555556). Square it (1.114197532) and add it to both sides. Add '1.114197532' to each side of the equation. -2.111111111r + 1.114197532 + r2 = -0.8888888889 + 1.114197532 Reorder the terms: 1.114197532 + -2.111111111r + r2 = -0.8888888889 + 1.114197532 Combine like terms: -0.8888888889 + 1.114197532 = 0.2253086431 1.114197532 + -2.111111111r + r2 = 0.2253086431 Factor a perfect square on the left side: (r + -1.055555556)(r + -1.055555556) = 0.2253086431 Calculate the square root of the right side: 0.474666876 Break this problem into two subproblems by setting (r + -1.055555556) equal to 0.474666876 and -0.474666876.Subproblem 1
r + -1.055555556 = 0.474666876 Simplifying r + -1.055555556 = 0.474666876 Reorder the terms: -1.055555556 + r = 0.474666876 Solving -1.055555556 + r = 0.474666876 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.055555556' to each side of the equation. -1.055555556 + 1.055555556 + r = 0.474666876 + 1.055555556 Combine like terms: -1.055555556 + 1.055555556 = 0.000000000 0.000000000 + r = 0.474666876 + 1.055555556 r = 0.474666876 + 1.055555556 Combine like terms: 0.474666876 + 1.055555556 = 1.530222432 r = 1.530222432 Simplifying r = 1.530222432Subproblem 2
r + -1.055555556 = -0.474666876 Simplifying r + -1.055555556 = -0.474666876 Reorder the terms: -1.055555556 + r = -0.474666876 Solving -1.055555556 + r = -0.474666876 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.055555556' to each side of the equation. -1.055555556 + 1.055555556 + r = -0.474666876 + 1.055555556 Combine like terms: -1.055555556 + 1.055555556 = 0.000000000 0.000000000 + r = -0.474666876 + 1.055555556 r = -0.474666876 + 1.055555556 Combine like terms: -0.474666876 + 1.055555556 = 0.58088868 r = 0.58088868 Simplifying r = 0.58088868Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.530222432, 0.58088868}
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