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Simplifying 3 = 10r2 + 10r Reorder the terms: 3 = 10r + 10r2 Solving 3 = 10r + 10r2 Solving for variable 'r'. Reorder the terms: 3 + -10r + -10r2 = 10r + -10r + 10r2 + -10r2 Combine like terms: 10r + -10r = 0 3 + -10r + -10r2 = 0 + 10r2 + -10r2 3 + -10r + -10r2 = 10r2 + -10r2 Combine like terms: 10r2 + -10r2 = 0 3 + -10r + -10r2 = 0 Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -0.3 + r + r2 = 0 Move the constant term to the right: Add '0.3' to each side of the equation. -0.3 + r + 0.3 + r2 = 0 + 0.3 Reorder the terms: -0.3 + 0.3 + r + r2 = 0 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + r + r2 = 0 + 0.3 r + r2 = 0 + 0.3 Combine like terms: 0 + 0.3 = 0.3 r + r2 = 0.3 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 0.3 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 0.3 + 0.25 Combine like terms: 0.3 + 0.25 = 0.55 1.25 + r2 = 0.55 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 0.55 Calculate the square root of the right side: 0.741619849 Break this problem into two subproblems by setting (r + 0.5) equal to 0.741619849 and -0.741619849.Subproblem 1
r + 0.5 = 0.741619849 Simplifying r + 0.5 = 0.741619849 Reorder the terms: 0.5 + r = 0.741619849 Solving 0.5 + r = 0.741619849 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 0.741619849 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.741619849 + -0.5 r = 0.741619849 + -0.5 Combine like terms: 0.741619849 + -0.5 = 0.241619849 r = 0.241619849 Simplifying r = 0.241619849Subproblem 2
r + 0.5 = -0.741619849 Simplifying r + 0.5 = -0.741619849 Reorder the terms: 0.5 + r = -0.741619849 Solving 0.5 + r = -0.741619849 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -0.741619849 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.741619849 + -0.5 r = -0.741619849 + -0.5 Combine like terms: -0.741619849 + -0.5 = -1.241619849 r = -1.241619849 Simplifying r = -1.241619849Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.241619849, -1.241619849}
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