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Simplifying (8r + -1)(5 + 2r) = (4r + 5) * 2 Reorder the terms: (-1 + 8r)(5 + 2r) = (4r + 5) * 2 Multiply (-1 + 8r) * (5 + 2r) (-1(5 + 2r) + 8r * (5 + 2r)) = (4r + 5) * 2 ((5 * -1 + 2r * -1) + 8r * (5 + 2r)) = (4r + 5) * 2 ((-5 + -2r) + 8r * (5 + 2r)) = (4r + 5) * 2 (-5 + -2r + (5 * 8r + 2r * 8r)) = (4r + 5) * 2 (-5 + -2r + (40r + 16r2)) = (4r + 5) * 2 Combine like terms: -2r + 40r = 38r (-5 + 38r + 16r2) = (4r + 5) * 2 Reorder the terms: -5 + 38r + 16r2 = (5 + 4r) * 2 Reorder the terms for easier multiplication: -5 + 38r + 16r2 = 2(5 + 4r) -5 + 38r + 16r2 = (5 * 2 + 4r * 2) -5 + 38r + 16r2 = (10 + 8r) Solving -5 + 38r + 16r2 = 10 + 8r Solving for variable 'r'. Reorder the terms: -5 + -10 + 38r + -8r + 16r2 = 10 + 8r + -10 + -8r Combine like terms: -5 + -10 = -15 -15 + 38r + -8r + 16r2 = 10 + 8r + -10 + -8r Combine like terms: 38r + -8r = 30r -15 + 30r + 16r2 = 10 + 8r + -10 + -8r Reorder the terms: -15 + 30r + 16r2 = 10 + -10 + 8r + -8r Combine like terms: 10 + -10 = 0 -15 + 30r + 16r2 = 0 + 8r + -8r -15 + 30r + 16r2 = 8r + -8r Combine like terms: 8r + -8r = 0 -15 + 30r + 16r2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.9375 + 1.875r + r2 = 0 Move the constant term to the right: Add '0.9375' to each side of the equation. -0.9375 + 1.875r + 0.9375 + r2 = 0 + 0.9375 Reorder the terms: -0.9375 + 0.9375 + 1.875r + r2 = 0 + 0.9375 Combine like terms: -0.9375 + 0.9375 = 0.0000 0.0000 + 1.875r + r2 = 0 + 0.9375 1.875r + r2 = 0 + 0.9375 Combine like terms: 0 + 0.9375 = 0.9375 1.875r + r2 = 0.9375 The r term is 1.875r. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875r + 0.87890625 + r2 = 0.9375 + 0.87890625 Reorder the terms: 0.87890625 + 1.875r + r2 = 0.9375 + 0.87890625 Combine like terms: 0.9375 + 0.87890625 = 1.81640625 0.87890625 + 1.875r + r2 = 1.81640625 Factor a perfect square on the left side: (r + 0.9375)(r + 0.9375) = 1.81640625 Calculate the square root of the right side: 1.347741166 Break this problem into two subproblems by setting (r + 0.9375) equal to 1.347741166 and -1.347741166.Subproblem 1
r + 0.9375 = 1.347741166 Simplifying r + 0.9375 = 1.347741166 Reorder the terms: 0.9375 + r = 1.347741166 Solving 0.9375 + r = 1.347741166 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + r = 1.347741166 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + r = 1.347741166 + -0.9375 r = 1.347741166 + -0.9375 Combine like terms: 1.347741166 + -0.9375 = 0.410241166 r = 0.410241166 Simplifying r = 0.410241166Subproblem 2
r + 0.9375 = -1.347741166 Simplifying r + 0.9375 = -1.347741166 Reorder the terms: 0.9375 + r = -1.347741166 Solving 0.9375 + r = -1.347741166 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + r = -1.347741166 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + r = -1.347741166 + -0.9375 r = -1.347741166 + -0.9375 Combine like terms: -1.347741166 + -0.9375 = -2.285241166 r = -2.285241166 Simplifying r = -2.285241166Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.410241166, -2.285241166}
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