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Simplifying 2r2 + -4r + 1 = 0 Reorder the terms: 1 + -4r + 2r2 = 0 Solving 1 + -4r + 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'. 0.5 + -2r + r2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2r + -0.5 + r2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2r + r2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2r + r2 = 0 + -0.5 -2r + r2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2r + r2 = -0.5 The r term is -2r. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2r + 1 + r2 = -0.5 + 1 Reorder the terms: 1 + -2r + r2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + -2r + r2 = 0.5 Factor a perfect square on the left side: (r + -1)(r + -1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (r + -1) equal to 0.707106781 and -0.707106781.Subproblem 1
r + -1 = 0.707106781 Simplifying r + -1 = 0.707106781 Reorder the terms: -1 + r = 0.707106781 Solving -1 + r = 0.707106781 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + r = 0.707106781 + 1 r = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 r = 1.707106781 Simplifying r = 1.707106781Subproblem 2
r + -1 = -0.707106781 Simplifying r + -1 = -0.707106781 Reorder the terms: -1 + r = -0.707106781 Solving -1 + r = -0.707106781 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + r = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + r = -0.707106781 + 1 r = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 r = 0.292893219 Simplifying r = 0.292893219Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.707106781, 0.292893219}
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