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Simplifying 6r2 + -9r + 2 = 0 Reorder the terms: 2 + -9r + 6r2 = 0 Solving 2 + -9r + 6r2 = 0 Solving for variable 'r'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.3333333333 + -1.5r + r2 = 0 Move the constant term to the right: Add '-0.3333333333' to each side of the equation. 0.3333333333 + -1.5r + -0.3333333333 + r2 = 0 + -0.3333333333 Reorder the terms: 0.3333333333 + -0.3333333333 + -1.5r + r2 = 0 + -0.3333333333 Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000 0.0000000000 + -1.5r + r2 = 0 + -0.3333333333 -1.5r + r2 = 0 + -0.3333333333 Combine like terms: 0 + -0.3333333333 = -0.3333333333 -1.5r + r2 = -0.3333333333 The r term is -1.5r. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5r + 0.5625 + r2 = -0.3333333333 + 0.5625 Reorder the terms: 0.5625 + -1.5r + r2 = -0.3333333333 + 0.5625 Combine like terms: -0.3333333333 + 0.5625 = 0.2291666667 0.5625 + -1.5r + r2 = 0.2291666667 Factor a perfect square on the left side: (r + -0.75)(r + -0.75) = 0.2291666667 Calculate the square root of the right side: 0.478713554 Break this problem into two subproblems by setting (r + -0.75) equal to 0.478713554 and -0.478713554.Subproblem 1
r + -0.75 = 0.478713554 Simplifying r + -0.75 = 0.478713554 Reorder the terms: -0.75 + r = 0.478713554 Solving -0.75 + r = 0.478713554 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = 0.478713554 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = 0.478713554 + 0.75 r = 0.478713554 + 0.75 Combine like terms: 0.478713554 + 0.75 = 1.228713554 r = 1.228713554 Simplifying r = 1.228713554Subproblem 2
r + -0.75 = -0.478713554 Simplifying r + -0.75 = -0.478713554 Reorder the terms: -0.75 + r = -0.478713554 Solving -0.75 + r = -0.478713554 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + r = -0.478713554 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + r = -0.478713554 + 0.75 r = -0.478713554 + 0.75 Combine like terms: -0.478713554 + 0.75 = 0.271286446 r = 0.271286446 Simplifying r = 0.271286446Solution
The solution to the problem is based on the solutions from the subproblems. r = {1.228713554, 0.271286446}
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