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Simplifying 75 = 0.05r(r + 2) Reorder the terms: 75 = 0.05r(2 + r) 75 = (2 * 0.05r + r * 0.05r) 75 = (0.1r + 0.05r2) Solving 75 = 0.1r + 0.05r2 Solving for variable 'r'. Reorder the terms: 75 + -0.1r + -0.05r2 = 0.1r + -0.1r + 0.05r2 + -0.05r2 Combine like terms: 0.1r + -0.1r = 0.0 75 + -0.1r + -0.05r2 = 0.0 + 0.05r2 + -0.05r2 75 + -0.1r + -0.05r2 = 0.05r2 + -0.05r2 Combine like terms: 0.05r2 + -0.05r2 = 0.00 75 + -0.1r + -0.05r2 = 0.00 Begin completing the square. Divide all terms by -0.05 the coefficient of the squared term: Divide each side by '-0.05'. -1500 + 2r + r2 = 0 Move the constant term to the right: Add '1500' to each side of the equation. -1500 + 2r + 1500 + r2 = 0 + 1500 Reorder the terms: -1500 + 1500 + 2r + r2 = 0 + 1500 Combine like terms: -1500 + 1500 = 0 0 + 2r + r2 = 0 + 1500 2r + r2 = 0 + 1500 Combine like terms: 0 + 1500 = 1500 2r + r2 = 1500 The r term is 2r. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2r + 1 + r2 = 1500 + 1 Reorder the terms: 1 + 2r + r2 = 1500 + 1 Combine like terms: 1500 + 1 = 1501 1 + 2r + r2 = 1501 Factor a perfect square on the left side: (r + 1)(r + 1) = 1501 Calculate the square root of the right side: 38.742741256 Break this problem into two subproblems by setting (r + 1) equal to 38.742741256 and -38.742741256.Subproblem 1
r + 1 = 38.742741256 Simplifying r + 1 = 38.742741256 Reorder the terms: 1 + r = 38.742741256 Solving 1 + r = 38.742741256 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = 38.742741256 + -1 Combine like terms: 1 + -1 = 0 0 + r = 38.742741256 + -1 r = 38.742741256 + -1 Combine like terms: 38.742741256 + -1 = 37.742741256 r = 37.742741256 Simplifying r = 37.742741256Subproblem 2
r + 1 = -38.742741256 Simplifying r + 1 = -38.742741256 Reorder the terms: 1 + r = -38.742741256 Solving 1 + r = -38.742741256 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + r = -38.742741256 + -1 Combine like terms: 1 + -1 = 0 0 + r = -38.742741256 + -1 r = -38.742741256 + -1 Combine like terms: -38.742741256 + -1 = -39.742741256 r = -39.742741256 Simplifying r = -39.742741256Solution
The solution to the problem is based on the solutions from the subproblems. r = {37.742741256, -39.742741256}
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