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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 = 0.75 2.25 + -3r + r2 = 0.75 Factor a perfect square on the left side: (r + -1.5)(r + -1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (r + -1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
r + -1.5 = 0.866025404 Simplifying r + -1.5 = 0.866025404 Reorder the terms: -1.5 + r = 0.866025404 Solving -1.5 + r = 0.866025404 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 = 0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = 0.866025404 + 1.5 r = 0.866025404 + 1.5 Combine like terms: 0.866025404 + 1.5 = 2.366025404 r = 2.366025404 Simplifying r = 2.366025404Subproblem 2
r + -1.5 = -0.866025404 Simplifying r + -1.5 = -0.866025404 Reorder the terms: -1.5 + r = -0.866025404 Solving -1.5 + r = -0.866025404 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 = -0.866025404 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + r = -0.866025404 + 1.5 r = -0.866025404 + 1.5 Combine like terms: -0.866025404 + 1.5 = 0.633974596 r = 0.633974596 Simplifying r = 0.633974596Solution
The solution to the problem is based on the solutions from the subproblems. r = {2.366025404, 0.633974596}
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