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Simplifying 2r2 + 3r + -3 = 0 Reorder the terms: -3 + 3r + 2r2 = 0 Solving -3 + 3r + 2r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 1.5r + r2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 1.5r + 1.5 + r2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 1.5r + r2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 1.5r + r2 = 0 + 1.5 1.5r + r2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 1.5r + r2 = 1.5 The r term is 1.5r. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5r + 0.5625 + r2 = 1.5 + 0.5625 Reorder the terms: 0.5625 + 1.5r + r2 = 1.5 + 0.5625 Combine like terms: 1.5 + 0.5625 = 2.0625 0.5625 + 1.5r + r2 = 2.0625 Factor a perfect square on the left side: (r + 0.75)(r + 0.75) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (r + 0.75) equal to 1.436140662 and -1.436140662.Subproblem 1
r + 0.75 = 1.436140662 Simplifying r + 0.75 = 1.436140662 Reorder the terms: 0.75 + r = 1.436140662 Solving 0.75 + r = 1.436140662 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = 1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = 1.436140662 + -0.75 r = 1.436140662 + -0.75 Combine like terms: 1.436140662 + -0.75 = 0.686140662 r = 0.686140662 Simplifying r = 0.686140662Subproblem 2
r + 0.75 = -1.436140662 Simplifying r + 0.75 = -1.436140662 Reorder the terms: 0.75 + r = -1.436140662 Solving 0.75 + r = -1.436140662 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + r = -1.436140662 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + r = -1.436140662 + -0.75 r = -1.436140662 + -0.75 Combine like terms: -1.436140662 + -0.75 = -2.186140662 r = -2.186140662 Simplifying r = -2.186140662Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.686140662, -2.186140662}
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