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Simplifying (4r + 2)(5r + 12) = 125 Reorder the terms: (2 + 4r)(5r + 12) = 125 Reorder the terms: (2 + 4r)(12 + 5r) = 125 Multiply (2 + 4r) * (12 + 5r) (2(12 + 5r) + 4r * (12 + 5r)) = 125 ((12 * 2 + 5r * 2) + 4r * (12 + 5r)) = 125 ((24 + 10r) + 4r * (12 + 5r)) = 125 (24 + 10r + (12 * 4r + 5r * 4r)) = 125 (24 + 10r + (48r + 20r2)) = 125 Combine like terms: 10r + 48r = 58r (24 + 58r + 20r2) = 125 Solving 24 + 58r + 20r2 = 125 Solving for variable 'r'. Reorder the terms: 24 + -125 + 58r + 20r2 = 125 + -125 Combine like terms: 24 + -125 = -101 -101 + 58r + 20r2 = 125 + -125 Combine like terms: 125 + -125 = 0 -101 + 58r + 20r2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -5.05 + 2.9r + r2 = 0.0 Move the constant term to the right: Add '5.05' to each side of the equation. -5.05 + 2.9r + 5.05 + r2 = 0.0 + 5.05 Reorder the terms: -5.05 + 5.05 + 2.9r + r2 = 0.0 + 5.05 Combine like terms: -5.05 + 5.05 = 0.00 0.00 + 2.9r + r2 = 0.0 + 5.05 2.9r + r2 = 0.0 + 5.05 Combine like terms: 0.0 + 5.05 = 5.05 2.9r + r2 = 5.05 The r term is 2.9r. Take half its coefficient (1.45). Square it (2.1025) and add it to both sides. Add '2.1025' to each side of the equation. 2.9r + 2.1025 + r2 = 5.05 + 2.1025 Reorder the terms: 2.1025 + 2.9r + r2 = 5.05 + 2.1025 Combine like terms: 5.05 + 2.1025 = 7.1525 2.1025 + 2.9r + r2 = 7.1525 Factor a perfect square on the left side: (r + 1.45)(r + 1.45) = 7.1525 Calculate the square root of the right side: 2.674415824 Break this problem into two subproblems by setting (r + 1.45) equal to 2.674415824 and -2.674415824.Subproblem 1
r + 1.45 = 2.674415824 Simplifying r + 1.45 = 2.674415824 Reorder the terms: 1.45 + r = 2.674415824 Solving 1.45 + r = 2.674415824 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.45' to each side of the equation. 1.45 + -1.45 + r = 2.674415824 + -1.45 Combine like terms: 1.45 + -1.45 = 0.00 0.00 + r = 2.674415824 + -1.45 r = 2.674415824 + -1.45 Combine like terms: 2.674415824 + -1.45 = 1.224415824 r = 1.224415824 Simplifying r = 1.224415824Subproblem 2
r + 1.45 = -2.674415824 Simplifying r + 1.45 = -2.674415824 Reorder the terms: 1.45 + r = -2.674415824 Solving 1.45 + r = -2.674415824 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1.45' to each side of the equation. 1.45 + -1.45 + r = -2.674415824 + -1.45 Combine like terms: 1.45 + -1.45 = 0.00 0.00 + r = -2.674415824 + -1.45 r = -2.674415824 + -1.45 Combine like terms: -2.674415824 + -1.45 = -4.124415824 r = -4.124415824 Simplifying r = -4.124415824Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.224415824, -4.124415824}
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