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Simplifying 6r2 + 6r + -1 = 0 Reorder the terms: -1 + 6r + 6r2 = 0 Solving -1 + 6r + 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'. -0.1666666667 + r + r2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + r + 0.1666666667 + r2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + r + r2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + r + r2 = 0 + 0.1666666667 r + r2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 r + r2 = 0.1666666667 The r term is r. 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. + 0.25 + r2 = 0.1666666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + r2 = 0.1666666667 + 0.25 Combine like terms: 0.1666666667 + 0.25 = 0.4166666667 1.25 + r2 = 0.4166666667 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 0.4166666667 Calculate the square root of the right side: 0.645497224 Break this problem into two subproblems by setting (r + 0.5) equal to 0.645497224 and -0.645497224.Subproblem 1
r + 0.5 = 0.645497224 Simplifying r + 0.5 = 0.645497224 Reorder the terms: 0.5 + r = 0.645497224 Solving 0.5 + r = 0.645497224 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 = 0.645497224 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 0.645497224 + -0.5 r = 0.645497224 + -0.5 Combine like terms: 0.645497224 + -0.5 = 0.145497224 r = 0.145497224 Simplifying r = 0.145497224Subproblem 2
r + 0.5 = -0.645497224 Simplifying r + 0.5 = -0.645497224 Reorder the terms: 0.5 + r = -0.645497224 Solving 0.5 + r = -0.645497224 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 = -0.645497224 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -0.645497224 + -0.5 r = -0.645497224 + -0.5 Combine like terms: -0.645497224 + -0.5 = -1.145497224 r = -1.145497224 Simplifying r = -1.145497224Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.145497224, -1.145497224}
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