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Simplifying 2r2 + 6r = 3 Reorder the terms: 6r + 2r2 = 3 Solving 6r + 2r2 = 3 Solving for variable 'r'. Reorder the terms: -3 + 6r + 2r2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 6r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 3r + r2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 3r + 1.5 + r2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 3r + r2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 3r + r2 = 0 + 1.5 3r + r2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 3r + r2 = 1.5 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 = 1.5 + 2.25 Reorder the terms: 2.25 + 3r + r2 = 1.5 + 2.25 Combine like terms: 1.5 + 2.25 = 3.75 2.25 + 3r + r2 = 3.75 Factor a perfect square on the left side: (r + 1.5)(r + 1.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (r + 1.5) equal to 1.936491673 and -1.936491673.Subproblem 1
r + 1.5 = 1.936491673 Simplifying r + 1.5 = 1.936491673 Reorder the terms: 1.5 + r = 1.936491673 Solving 1.5 + r = 1.936491673 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 = 1.936491673 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = 1.936491673 + -1.5 r = 1.936491673 + -1.5 Combine like terms: 1.936491673 + -1.5 = 0.436491673 r = 0.436491673 Simplifying r = 0.436491673Subproblem 2
r + 1.5 = -1.936491673 Simplifying r + 1.5 = -1.936491673 Reorder the terms: 1.5 + r = -1.936491673 Solving 1.5 + r = -1.936491673 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 = -1.936491673 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + r = -1.936491673 + -1.5 r = -1.936491673 + -1.5 Combine like terms: -1.936491673 + -1.5 = -3.436491673 r = -3.436491673 Simplifying r = -3.436491673Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.436491673, -3.436491673}
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