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Simplifying (0.6r + -0.48)(0.05r + 1) = 0 Reorder the terms: (-0.48 + 0.6r)(0.05r + 1) = 0 Reorder the terms: (-0.48 + 0.6r)(1 + 0.05r) = 0 Multiply (-0.48 + 0.6r) * (1 + 0.05r) (-0.48(1 + 0.05r) + 0.6r * (1 + 0.05r)) = 0 ((1 * -0.48 + 0.05r * -0.48) + 0.6r * (1 + 0.05r)) = 0 ((-0.48 + -0.024r) + 0.6r * (1 + 0.05r)) = 0 (-0.48 + -0.024r + (1 * 0.6r + 0.05r * 0.6r)) = 0 (-0.48 + -0.024r + (0.6r + 0.03r2)) = 0 Combine like terms: -0.024r + 0.6r = 0.576r (-0.48 + 0.576r + 0.03r2) = 0 Solving -0.48 + 0.576r + 0.03r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 0.03 the coefficient of the squared term: Divide each side by '0.03'. -16 + 19.2r + r2 = 0 Move the constant term to the right: Add '16' to each side of the equation. -16 + 19.2r + 16 + r2 = 0 + 16 Reorder the terms: -16 + 16 + 19.2r + r2 = 0 + 16 Combine like terms: -16 + 16 = 0 0 + 19.2r + r2 = 0 + 16 19.2r + r2 = 0 + 16 Combine like terms: 0 + 16 = 16 19.2r + r2 = 16 The r term is 19.2r. Take half its coefficient (9.6). Square it (92.16) and add it to both sides. Add '92.16' to each side of the equation. 19.2r + 92.16 + r2 = 16 + 92.16 Reorder the terms: 92.16 + 19.2r + r2 = 16 + 92.16 Combine like terms: 16 + 92.16 = 108.16 92.16 + 19.2r + r2 = 108.16 Factor a perfect square on the left side: (r + 9.6)(r + 9.6) = 108.16 Calculate the square root of the right side: 10.4 Break this problem into two subproblems by setting (r + 9.6) equal to 10.4 and -10.4.Subproblem 1
r + 9.6 = 10.4 Simplifying r + 9.6 = 10.4 Reorder the terms: 9.6 + r = 10.4 Solving 9.6 + r = 10.4 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-9.6' to each side of the equation. 9.6 + -9.6 + r = 10.4 + -9.6 Combine like terms: 9.6 + -9.6 = 0.0 0.0 + r = 10.4 + -9.6 r = 10.4 + -9.6 Combine like terms: 10.4 + -9.6 = 0.8 r = 0.8 Simplifying r = 0.8Subproblem 2
r + 9.6 = -10.4 Simplifying r + 9.6 = -10.4 Reorder the terms: 9.6 + r = -10.4 Solving 9.6 + r = -10.4 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-9.6' to each side of the equation. 9.6 + -9.6 + r = -10.4 + -9.6 Combine like terms: 9.6 + -9.6 = 0.0 0.0 + r = -10.4 + -9.6 r = -10.4 + -9.6 Combine like terms: -10.4 + -9.6 = -20 r = -20 Simplifying r = -20Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.8, -20}
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