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Simplifying -2r2 + -1r + 1 = r Reorder the terms: 1 + -1r + -2r2 = r Solving 1 + -1r + -2r2 = r Solving for variable 'r'. Reorder the terms: 1 + -1r + -1r + -2r2 = r + -1r Combine like terms: -1r + -1r = -2r 1 + -2r + -2r2 = r + -1r Combine like terms: r + -1r = 0 1 + -2r + -2r2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + r + r2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + r + 0.5 + r2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + r + r2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r + r2 = 0 + 0.5 r + r2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 r + r2 = 0.5 The r term is r. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + r2 = 0.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 1.25 + r2 = 0.75 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (r + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
r + 0.5 = 0.866025404 Simplifying r + 0.5 = 0.866025404 Reorder the terms: 0.5 + r = 0.866025404 Solving 0.5 + r = 0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.866025404 + -0.5 r = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 r = 0.366025404 Simplifying r = 0.366025404Subproblem 2
r + 0.5 = -0.866025404 Simplifying r + 0.5 = -0.866025404 Reorder the terms: 0.5 + r = -0.866025404 Solving 0.5 + r = -0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + r = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.866025404 + -0.5 r = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 r = -1.366025404 Simplifying r = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.366025404, -1.366025404}
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