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Simplifying 2r2 + -3r = 7 Reorder the terms: -3r + 2r2 = 7 Solving -3r + 2r2 = 7 Solving for variable 'r'. Reorder the terms: -7 + -3r + 2r2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + -3r + 2r2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -1.5r + r2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -1.5r + 3.5 + r2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -1.5r + r2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -1.5r + r2 = 0 + 3.5 -1.5r + r2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -1.5r + r2 = 3.5 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 = 3.5 + 0.5625 Reorder the terms: 0.5625 + -1.5r + r2 = 3.5 + 0.5625 Combine like terms: 3.5 + 0.5625 = 4.0625 0.5625 + -1.5r + r2 = 4.0625 Factor a perfect square on the left side: (r + -0.75)(r + -0.75) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (r + -0.75) equal to 2.015564437 and -2.015564437.Subproblem 1
r + -0.75 = 2.015564437 Simplifying r + -0.75 = 2.015564437 Reorder the terms: -0.75 + r = 2.015564437 Solving -0.75 + r = 2.015564437 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 = 2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 2.015564437 + 0.75 r = 2.015564437 + 0.75 Combine like terms: 2.015564437 + 0.75 = 2.765564437 r = 2.765564437 Simplifying r = 2.765564437Subproblem 2
r + -0.75 = -2.015564437 Simplifying r + -0.75 = -2.015564437 Reorder the terms: -0.75 + r = -2.015564437 Solving -0.75 + r = -2.015564437 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 = -2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -2.015564437 + 0.75 r = -2.015564437 + 0.75 Combine like terms: -2.015564437 + 0.75 = -1.265564437 r = -1.265564437 Simplifying r = -1.265564437Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.765564437, -1.265564437}
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