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Simplifying 3r + 5 = r2 Reorder the terms: 5 + 3r = r2 Solving 5 + 3r = r2 Solving for variable 'r'. Combine like terms: r2 + -1r2 = 0 5 + 3r + -1r2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -5 + -3r + r2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -3r + 5 + r2 = 0 + 5 Reorder the terms: -5 + 5 + -3r + r2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -3r + r2 = 0 + 5 -3r + r2 = 0 + 5 Combine like terms: 0 + 5 = 5 -3r + r2 = 5 The r term is -3r. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3r + 2.25 + r2 = 5 + 2.25 Reorder the terms: 2.25 + -3r + r2 = 5 + 2.25 Combine like terms: 5 + 2.25 = 7.25 2.25 + -3r + r2 = 7.25 Factor a perfect square on the left side: (r + -1.5)(r + -1.5) = 7.25 Calculate the square root of the right side: 2.692582404 Break this problem into two subproblems by setting (r + -1.5) equal to 2.692582404 and -2.692582404.Subproblem 1
r + -1.5 = 2.692582404 Simplifying r + -1.5 = 2.692582404 Reorder the terms: -1.5 + r = 2.692582404 Solving -1.5 + r = 2.692582404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = 2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = 2.692582404 + 1.5 r = 2.692582404 + 1.5 Combine like terms: 2.692582404 + 1.5 = 4.192582404 r = 4.192582404 Simplifying r = 4.192582404Subproblem 2
r + -1.5 = -2.692582404 Simplifying r + -1.5 = -2.692582404 Reorder the terms: -1.5 + r = -2.692582404 Solving -1.5 + r = -2.692582404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + r = -2.692582404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = -2.692582404 + 1.5 r = -2.692582404 + 1.5 Combine like terms: -2.692582404 + 1.5 = -1.192582404 r = -1.192582404 Simplifying r = -1.192582404Solution
The solution to the problem is based on the solutions from the subproblems. r = {4.192582404, -1.192582404}
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