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Simplifying 3(10r + 1) * r = 2 Reorder the terms: 3(1 + 10r) * r = 2 Reorder the terms for easier multiplication: 3r(1 + 10r) = 2 (1 * 3r + 10r * 3r) = 2 (3r + 30r2) = 2 Solving 3r + 30r2 = 2 Solving for variable 'r'. Reorder the terms: -2 + 3r + 30r2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + 3r + 30r2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -0.06666666667 + 0.1r + r2 = 0.0 Move the constant term to the right: Add '0.06666666667' to each side of the equation. -0.06666666667 + 0.1r + 0.06666666667 + r2 = 0.0 + 0.06666666667 Reorder the terms: -0.06666666667 + 0.06666666667 + 0.1r + r2 = 0.0 + 0.06666666667 Combine like terms: -0.06666666667 + 0.06666666667 = 0.00000000000 0.00000000000 + 0.1r + r2 = 0.0 + 0.06666666667 0.1r + r2 = 0.0 + 0.06666666667 Combine like terms: 0.0 + 0.06666666667 = 0.06666666667 0.1r + r2 = 0.06666666667 The r term is 0.1r. Take half its coefficient (0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. 0.1r + 0.0025 + r2 = 0.06666666667 + 0.0025 Reorder the terms: 0.0025 + 0.1r + r2 = 0.06666666667 + 0.0025 Combine like terms: 0.06666666667 + 0.0025 = 0.06916666667 0.0025 + 0.1r + r2 = 0.06916666667 Factor a perfect square on the left side: (r + 0.05)(r + 0.05) = 0.06916666667 Calculate the square root of the right side: 0.262995564 Break this problem into two subproblems by setting (r + 0.05) equal to 0.262995564 and -0.262995564.Subproblem 1
r + 0.05 = 0.262995564 Simplifying r + 0.05 = 0.262995564 Reorder the terms: 0.05 + r = 0.262995564 Solving 0.05 + r = 0.262995564 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.05' to each side of the equation. 0.05 + -0.05 + r = 0.262995564 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + r = 0.262995564 + -0.05 r = 0.262995564 + -0.05 Combine like terms: 0.262995564 + -0.05 = 0.212995564 r = 0.212995564 Simplifying r = 0.212995564Subproblem 2
r + 0.05 = -0.262995564 Simplifying r + 0.05 = -0.262995564 Reorder the terms: 0.05 + r = -0.262995564 Solving 0.05 + r = -0.262995564 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.05' to each side of the equation. 0.05 + -0.05 + r = -0.262995564 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + r = -0.262995564 + -0.05 r = -0.262995564 + -0.05 Combine like terms: -0.262995564 + -0.05 = -0.312995564 r = -0.312995564 Simplifying r = -0.312995564Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.212995564, -0.312995564}
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