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Simplifying r + -6r2 = -11 Solving r + -6r2 = -11 Solving for variable 'r'. Reorder the terms: 11 + r + -6r2 = -11 + 11 Combine like terms: -11 + 11 = 0 11 + r + -6r2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.833333333 + -0.1666666667r + r2 = 0 Move the constant term to the right: Add '1.833333333' to each side of the equation. -1.833333333 + -0.1666666667r + 1.833333333 + r2 = 0 + 1.833333333 Reorder the terms: -1.833333333 + 1.833333333 + -0.1666666667r + r2 = 0 + 1.833333333 Combine like terms: -1.833333333 + 1.833333333 = 0.000000000 0.000000000 + -0.1666666667r + r2 = 0 + 1.833333333 -0.1666666667r + r2 = 0 + 1.833333333 Combine like terms: 0 + 1.833333333 = 1.833333333 -0.1666666667r + r2 = 1.833333333 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. -0.1666666667r + 0.25 + r2 = 1.833333333 + 0.25 Reorder the terms: 0.25 + -0.1666666667r + r2 = 1.833333333 + 0.25 Combine like terms: 1.833333333 + 0.25 = 2.083333333 0.25 + -0.1666666667r + r2 = 2.083333333 Factor a perfect square on the left side: (r + 0.5)(r + 0.5) = 2.083333333 Calculate the square root of the right side: 1.443375673 Break this problem into two subproblems by setting (r + 0.5) equal to 1.443375673 and -1.443375673.Subproblem 1
r + 0.5 = 1.443375673 Simplifying r + 0.5 = 1.443375673 Reorder the terms: 0.5 + r = 1.443375673 Solving 0.5 + r = 1.443375673 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 = 1.443375673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = 1.443375673 + -0.5 r = 1.443375673 + -0.5 Combine like terms: 1.443375673 + -0.5 = 0.943375673 r = 0.943375673 Simplifying r = 0.943375673Subproblem 2
r + 0.5 = -1.443375673 Simplifying r + 0.5 = -1.443375673 Reorder the terms: 0.5 + r = -1.443375673 Solving 0.5 + r = -1.443375673 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 = -1.443375673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + r = -1.443375673 + -0.5 r = -1.443375673 + -0.5 Combine like terms: -1.443375673 + -0.5 = -1.943375673 r = -1.943375673 Simplifying r = -1.943375673Solution
The solution to the problem is based on the solutions from the subproblems. r = {0.943375673, -1.943375673}
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