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Simplifying (0.6r + -0.1)(0.11r + 1) = 0 Reorder the terms: (-0.1 + 0.6r)(0.11r + 1) = 0 Reorder the terms: (-0.1 + 0.6r)(1 + 0.11r) = 0 Multiply (-0.1 + 0.6r) * (1 + 0.11r) (-0.1(1 + 0.11r) + 0.6r * (1 + 0.11r)) = 0 ((1 * -0.1 + 0.11r * -0.1) + 0.6r * (1 + 0.11r)) = 0 ((-0.1 + -0.011r) + 0.6r * (1 + 0.11r)) = 0 (-0.1 + -0.011r + (1 * 0.6r + 0.11r * 0.6r)) = 0 (-0.1 + -0.011r + (0.6r + 0.066r2)) = 0 Combine like terms: -0.011r + 0.6r = 0.589r (-0.1 + 0.589r + 0.066r2) = 0 Solving -0.1 + 0.589r + 0.066r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 0.066 the coefficient of the squared term: Divide each side by '0.066'. -1.515151515 + 8.924242424r + r2 = 0 Move the constant term to the right: Add '1.515151515' to each side of the equation. -1.515151515 + 8.924242424r + 1.515151515 + r2 = 0 + 1.515151515 Reorder the terms: -1.515151515 + 1.515151515 + 8.924242424r + r2 = 0 + 1.515151515 Combine like terms: -1.515151515 + 1.515151515 = 0.000000000 0.000000000 + 8.924242424r + r2 = 0 + 1.515151515 8.924242424r + r2 = 0 + 1.515151515 Combine like terms: 0 + 1.515151515 = 1.515151515 8.924242424r + r2 = 1.515151515 The r term is 8.924242424r. Take half its coefficient (4.462121212). Square it (19.91052571) and add it to both sides. Add '19.91052571' to each side of the equation. 8.924242424r + 19.91052571 + r2 = 1.515151515 + 19.91052571 Reorder the terms: 19.91052571 + 8.924242424r + r2 = 1.515151515 + 19.91052571 Combine like terms: 1.515151515 + 19.91052571 = 21.425677225 19.91052571 + 8.924242424r + r2 = 21.425677225 Factor a perfect square on the left side: (r + 4.462121212)(r + 4.462121212) = 21.425677225 Calculate the square root of the right side: 4.628787879 Break this problem into two subproblems by setting (r + 4.462121212) equal to 4.628787879 and -4.628787879.Subproblem 1
r + 4.462121212 = 4.628787879 Simplifying r + 4.462121212 = 4.628787879 Reorder the terms: 4.462121212 + r = 4.628787879 Solving 4.462121212 + r = 4.628787879 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.462121212' to each side of the equation. 4.462121212 + -4.462121212 + r = 4.628787879 + -4.462121212 Combine like terms: 4.462121212 + -4.462121212 = 0.000000000 0.000000000 + r = 4.628787879 + -4.462121212 r = 4.628787879 + -4.462121212 Combine like terms: 4.628787879 + -4.462121212 = 0.166666667 r = 0.166666667 Simplifying r = 0.166666667Subproblem 2
r + 4.462121212 = -4.628787879 Simplifying r + 4.462121212 = -4.628787879 Reorder the terms: 4.462121212 + r = -4.628787879 Solving 4.462121212 + r = -4.628787879 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-4.462121212' to each side of the equation. 4.462121212 + -4.462121212 + r = -4.628787879 + -4.462121212 Combine like terms: 4.462121212 + -4.462121212 = 0.000000000 0.000000000 + r = -4.628787879 + -4.462121212 r = -4.628787879 + -4.462121212 Combine like terms: -4.628787879 + -4.462121212 = -9.090909091 r = -9.090909091 Simplifying r = -9.090909091Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.166666667, -9.090909091}
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