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Simplifying 4r2 = 8r + -1 Reorder the terms: 4r2 = -1 + 8r Solving 4r2 = -1 + 8r Solving for variable 'r'. Reorder the terms: 1 + -8r + 4r2 = -1 + 8r + 1 + -8r Reorder the terms: 1 + -8r + 4r2 = -1 + 1 + 8r + -8r Combine like terms: -1 + 1 = 0 1 + -8r + 4r2 = 0 + 8r + -8r 1 + -8r + 4r2 = 8r + -8r Combine like terms: 8r + -8r = 0 1 + -8r + 4r2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -2r + r2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -2r + -0.25 + r2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -2r + r2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -2r + r2 = 0 + -0.25 -2r + r2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -2r + r2 = -0.25 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 = -0.25 + 1 Reorder the terms: 1 + -2r + r2 = -0.25 + 1 Combine like terms: -0.25 + 1 = 0.75 1 + -2r + r2 = 0.75 Factor a perfect square on the left side: (r + -1)(r + -1) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (r + -1) equal to 0.866025404 and -0.866025404.Subproblem 1
r + -1 = 0.866025404 Simplifying r + -1 = 0.866025404 Reorder the terms: -1 + r = 0.866025404 Solving -1 + r = 0.866025404 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 = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + r = 0.866025404 + 1 r = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 r = 1.866025404 Simplifying r = 1.866025404Subproblem 2
r + -1 = -0.866025404 Simplifying r + -1 = -0.866025404 Reorder the terms: -1 + r = -0.866025404 Solving -1 + r = -0.866025404 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 = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + r = -0.866025404 + 1 r = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 r = 0.133974596 Simplifying r = 0.133974596Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.866025404, 0.133974596}
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