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Simplifying 20r2 = 20r + 1 Reorder the terms: 20r2 = 1 + 20r Solving 20r2 = 1 + 20r Solving for variable 'r'. Reorder the terms: -1 + -20r + 20r2 = 1 + 20r + -1 + -20r Reorder the terms: -1 + -20r + 20r2 = 1 + -1 + 20r + -20r Combine like terms: 1 + -1 = 0 -1 + -20r + 20r2 = 0 + 20r + -20r -1 + -20r + 20r2 = 20r + -20r Combine like terms: 20r + -20r = 0 -1 + -20r + 20r2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.05 + -1r + r2 = 0 Move the constant term to the right: Add '0.05' to each side of the equation. -0.05 + -1r + 0.05 + r2 = 0 + 0.05 Reorder the terms: -0.05 + 0.05 + -1r + r2 = 0 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + -1r + r2 = 0 + 0.05 -1r + r2 = 0 + 0.05 Combine like terms: 0 + 0.05 = 0.05 -1r + r2 = 0.05 The r term is -1r. 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. -1r + 0.25 + r2 = 0.05 + 0.25 Reorder the terms: 0.25 + -1r + r2 = 0.05 + 0.25 Combine like terms: 0.05 + 0.25 = 0.3 0.25 + -1r + r2 = 0.3 Factor a perfect square on the left side: (r + -0.5)(r + -0.5) = 0.3 Calculate the square root of the right side: 0.547722558 Break this problem into two subproblems by setting (r + -0.5) equal to 0.547722558 and -0.547722558.Subproblem 1
r + -0.5 = 0.547722558 Simplifying r + -0.5 = 0.547722558 Reorder the terms: -0.5 + r = 0.547722558 Solving -0.5 + r = 0.547722558 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 = 0.547722558 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = 0.547722558 + 0.5 r = 0.547722558 + 0.5 Combine like terms: 0.547722558 + 0.5 = 1.047722558 r = 1.047722558 Simplifying r = 1.047722558Subproblem 2
r + -0.5 = -0.547722558 Simplifying r + -0.5 = -0.547722558 Reorder the terms: -0.5 + r = -0.547722558 Solving -0.5 + r = -0.547722558 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 = -0.547722558 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + r = -0.547722558 + 0.5 r = -0.547722558 + 0.5 Combine like terms: -0.547722558 + 0.5 = -0.047722558 r = -0.047722558 Simplifying r = -0.047722558Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.047722558, -0.047722558}
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