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Simplifying 9r2 + -11r + 3 = 0 Reorder the terms: 3 + -11r + 9r2 = 0 Solving 3 + -11r + 9r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 0.3333333333 + -1.222222222r + r2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -1.222222222r + -0.3333333333 + r2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -1.222222222r + r2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -1.222222222r + r2 = 0 + -0.3333333333 -1.222222222r + r2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -1.222222222r + r2 = -0.3333333333 The r term is -1.222222222r. Take half its coefficient (-0.611111111). Square it (0.3734567900) and add it to both sides. Add '0.3734567900' to each side of the equation. -1.222222222r + 0.3734567900 + r2 = -0.3333333333 + 0.3734567900 Reorder the terms: 0.3734567900 + -1.222222222r + r2 = -0.3333333333 + 0.3734567900 Combine like terms: -0.3333333333 + 0.3734567900 = 0.0401234567 0.3734567900 + -1.222222222r + r2 = 0.0401234567 Factor a perfect square on the left side: (r + -0.611111111)(r + -0.611111111) = 0.0401234567 Calculate the square root of the right side: 0.200308404 Break this problem into two subproblems by setting (r + -0.611111111) equal to 0.200308404 and -0.200308404.Subproblem 1
r + -0.611111111 = 0.200308404 Simplifying r + -0.611111111 = 0.200308404 Reorder the terms: -0.611111111 + r = 0.200308404 Solving -0.611111111 + r = 0.200308404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + r = 0.200308404 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + r = 0.200308404 + 0.611111111 r = 0.200308404 + 0.611111111 Combine like terms: 0.200308404 + 0.611111111 = 0.811419515 r = 0.811419515 Simplifying r = 0.811419515Subproblem 2
r + -0.611111111 = -0.200308404 Simplifying r + -0.611111111 = -0.200308404 Reorder the terms: -0.611111111 + r = -0.200308404 Solving -0.611111111 + r = -0.200308404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.611111111' to each side of the equation. -0.611111111 + 0.611111111 + r = -0.200308404 + 0.611111111 Combine like terms: -0.611111111 + 0.611111111 = 0.000000000 0.000000000 + r = -0.200308404 + 0.611111111 r = -0.200308404 + 0.611111111 Combine like terms: -0.200308404 + 0.611111111 = 0.410802707 r = 0.410802707 Simplifying r = 0.410802707Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.811419515, 0.410802707}
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