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Simplifying (0.3r + 0.3)(r + 0.7) = 0 Reorder the terms: (0.3 + 0.3r)(r + 0.7) = 0 Reorder the terms: (0.3 + 0.3r)(0.7 + r) = 0 Multiply (0.3 + 0.3r) * (0.7 + r) (0.3(0.7 + r) + 0.3r * (0.7 + r)) = 0 ((0.7 * 0.3 + r * 0.3) + 0.3r * (0.7 + r)) = 0 ((0.21 + 0.3r) + 0.3r * (0.7 + r)) = 0 (0.21 + 0.3r + (0.7 * 0.3r + r * 0.3r)) = 0 (0.21 + 0.3r + (0.21r + 0.3r2)) = 0 Combine like terms: 0.3r + 0.21r = 0.51r (0.21 + 0.51r + 0.3r2) = 0 Solving 0.21 + 0.51r + 0.3r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 0.3 the coefficient of the squared term: Divide each side by '0.3'. 0.7 + 1.7r + r2 = 0 Move the constant term to the right: Add '-0.7' to each side of the equation. 0.7 + 1.7r + -0.7 + r2 = 0 + -0.7 Reorder the terms: 0.7 + -0.7 + 1.7r + r2 = 0 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + 1.7r + r2 = 0 + -0.7 1.7r + r2 = 0 + -0.7 Combine like terms: 0 + -0.7 = -0.7 1.7r + r2 = -0.7 The r term is 1.7r. Take half its coefficient (0.85). Square it (0.7225) and add it to both sides. Add '0.7225' to each side of the equation. 1.7r + 0.7225 + r2 = -0.7 + 0.7225 Reorder the terms: 0.7225 + 1.7r + r2 = -0.7 + 0.7225 Combine like terms: -0.7 + 0.7225 = 0.0225 0.7225 + 1.7r + r2 = 0.0225 Factor a perfect square on the left side: (r + 0.85)(r + 0.85) = 0.0225 Calculate the square root of the right side: 0.15 Break this problem into two subproblems by setting (r + 0.85) equal to 0.15 and -0.15.Subproblem 1
r + 0.85 = 0.15 Simplifying r + 0.85 = 0.15 Reorder the terms: 0.85 + r = 0.15 Solving 0.85 + r = 0.15 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.85' to each side of the equation. 0.85 + -0.85 + r = 0.15 + -0.85 Combine like terms: 0.85 + -0.85 = 0.00 0.00 + r = 0.15 + -0.85 r = 0.15 + -0.85 Combine like terms: 0.15 + -0.85 = -0.7 r = -0.7 Simplifying r = -0.7Subproblem 2
r + 0.85 = -0.15 Simplifying r + 0.85 = -0.15 Reorder the terms: 0.85 + r = -0.15 Solving 0.85 + r = -0.15 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.85' to each side of the equation. 0.85 + -0.85 + r = -0.15 + -0.85 Combine like terms: 0.85 + -0.85 = 0.00 0.00 + r = -0.15 + -0.85 r = -0.15 + -0.85 Combine like terms: -0.15 + -0.85 = -1 r = -1 Simplifying r = -1Solution
The solution to the problem is based on the solutions from the subproblems. r = {-0.7, -1}
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